From 7b4f4ee4813f8afc3e9949a368fe708351d3b89c Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 26 May 2017 10:20:59 -0400 Subject: changes --- .gitignore | 5 + essential_ising.bib | 99 ++++++++++++++++++ essential_ising.tex | 206 ++++++++++++++++++++++++++++++++++++++ figs/scaling_func.gplot | 16 +++ figs/scaling_func.tex | 260 ++++++++++++++++++++++++++++++++++++++++++++++++ makefile | 13 +++ 6 files changed, 599 insertions(+) create mode 100644 .gitignore create mode 100644 essential_ising.bib create mode 100644 essential_ising.tex create mode 100644 figs/scaling_func.gplot create mode 100644 figs/scaling_func.tex create mode 100644 makefile diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..6e3b920 --- /dev/null +++ b/.gitignore @@ -0,0 +1,5 @@ +*.pdf +*.aux +*.log +*.bbl +*.blg diff --git a/essential_ising.bib b/essential_ising.bib new file mode 100644 index 0000000..c84c1a7 --- /dev/null +++ b/essential_ising.bib @@ -0,0 +1,99 @@ + +@article{houghton.1980.metastable, + title={The metastable Ising magnet in a negative field}, + author={Hougton, Anthony and Lubensky, Tom C}, + journal={Physics Letters A}, + volume={77}, + number={6}, + pages={479--480}, + year={1980}, + publisher={North-Holland} +} + +@article{langer.1967.condensation, + title={Theory of the condensation point}, + author={Langer, James S}, + journal={Annals of Physics}, + volume={41}, + number={1}, + pages={108--157}, + year={1967}, + publisher={Elsevier} +} + +@article{langer.1969.metastable, + title={Statistical theory of the decay of metastable states}, + author={Langer, JS}, + journal={Annals of Physics}, + volume={54}, + number={2}, + pages={258--275}, + year={1969}, + publisher={Elsevier} +} + +@article{gaveau.1989.analytic, + title={Metastable decay rates and analytic continuation}, + author={Gaveau, B and Schulman, LS}, + journal={Letters in Mathematical Physics}, + volume={18}, + number={3}, + pages={201--208}, + year={1989}, + publisher={Springer} +} + +@article{bogomolny.1977.dispersion, + title={Calculation of the green functions by the coupling constant dispersion relations}, + author={Bogomolny, Evgeny B}, + journal={Physics Letters B}, + volume={67}, + number={2}, + pages={193--194}, + year={1977}, + publisher={Elsevier} +} + +@article{gunther.1993.transfer-matrix, + title={Numerical transfer-matrix study of metastability in the d= 2 Ising model}, + author={G{\"u}nther, Christoph CA and Rikvold, Per Arne and Novotny, MA}, + journal={Physical review letters}, + volume={71}, + number={24}, + pages={3898}, + year={1993}, + publisher={APS} +} + +@inproceedings{widom.1981.interface, + title={Structure of the interface between fluid phases}, + author={Widom, Benjamin}, + booktitle={Faraday Symposia of the Chemical Society}, + volume={16}, + pages={7--21}, + year={1981}, + organization={Royal Society of Chemistry} +} + +@article{gunther.1980.goldstone, + title={Goldstone modes in vacuum decay and first-order phase transitions}, + author={G{\"u}nther, NJ and Wallace, DJ and Nicole, DA}, + journal={Journal of Physics A: Mathematical and General}, + volume={13}, + number={5}, + pages={1755}, + year={1980}, + publisher={IOP Publishing} +} + +@article{barouch.1973.susceptibility, + title={Zero-field susceptibility of the two-dimensional Ising model near ${T}_c$}, + author={Barouch, Eytan and McCoy, Barry M and Wu, Tai Tsun}, + journal={Physical Review Letters}, + volume={31}, + number={23}, + pages={1409}, + year={1973}, + publisher={APS} +} + diff --git a/essential_ising.tex b/essential_ising.tex new file mode 100644 index 0000000..d6a92cc --- /dev/null +++ b/essential_ising.tex @@ -0,0 +1,206 @@ +% Ising model abrupt transition. +% +% Created by Jaron Kent-Dobias on Thu Apr 20 12:50:56 EDT 2017. +% Copyright (c) 2017 Jaron Kent-Dobias. All rights reserved. +% +\documentclass[fleqn]{article} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{amsmath,amssymb,latexsym,concmath,mathtools,xifthen,mfpic} + +\mathtoolsset{showonlyrefs=true} + +\title{Essential Singularity in the Ising Abrupt Transition} +\author{Jaron Kent-Dobias} + +\date{April 20, 2017} + +\begin{document} + +\def\[{\begin{equation}} +\def\]{\end{equation}} + +\def\im{\mathop{\mathrm{Im}}\nolimits} +\def\dd{\mathrm d} +\def\O{\mathcal O} +\def\ei{\mathop{\mathrm{Ei}}\nolimits} +\def\b{\mathrm b} + +\newcommand\pd[3][]{ + \ifthenelse{\isempty{#1}} + {\def\tmp{}} + {\def\tmp{^#1}} + \frac{\partial\tmp#2}{\partial#3\tmp} +} + +\maketitle + +\begin{abstract} +\end{abstract} + +It's long been known that the decay rate $\Gamma$ of metastable states in +statistical mechanics is often related to the metastable free energy $F$ by +\cite{langer.1967.condensation,langer.1969.metastable,gaveau.1989.analytic} +\[ + \Gamma\propto\im F +\] +What exactly is meant by `metastable free energy' is important to establish, +since formally the free energy relies on the existence of an equilibrium +state. Here one can imagine either analytic continuation of the free energy +through an abrupt phase transition, or restriction of the partition function +trace to states in the vicinity of the local free energy minimum that +characterizes the metastable state. In any case, the free energy develops a +nonzero imaginary part in the metastable region. Heuristically, this can be +thought of as similar to what happens in quantum mechanics with a non-unitary +Hamiltonian: the imaginary part describes loss of probability in the system +that corresponds to decay. + +One can estimate the scaling of the decay rate of the {\sc 2d} Ising model +using ideas from nucleation theory. In this framework, the metastable state +decays when a sufficiently large domain in the stable state forms to grow +stably to fill out the whole system. The free energy of a domain of $N$ spins +causes a free energy change +\[ + \Delta F=\Sigma N^\sigma-MHN +\] +where $\Sigma$ is the surface tension and $1-\frac1d\leq\sigma<1$. This is +maximized by +\[ + N_c=\bigg(\frac{MH}{\sigma\Sigma}\bigg)^{-1/(\sigma-1)} +\] +which corresponds to a free energy change +\[ + \Delta F_c\sim\bigg(\frac\Sigma{(MH)^\sigma}\bigg)^{1/(1-\sigma)} +\] +The rate of formation is proportional to the Boltzmann factor, +\[ + \Gamma\sim e^{-\beta \Delta + F_c}=e^{-\beta(\Sigma/(MH)^\sigma)^{1/(1-\sigma)}} +\] +For domains whose boundary is minimal, $\sigma=1-\frac1d$ and this becomes +\[ + \Gamma\sim e^{-\beta(\Sigma/(MH)^\sigma)^{d-1}} +\] +Since $\Sigma\sim t^\mu\mathcal S(ht^{-\beta\delta})$ with $\mu=-\nu+\gamma+2\beta$ +\cite{widom.1981.interface} and $M\sim t^\beta\mathcal M(ht^{-\beta\delta})$ +with $\mathcal S(0)=\O(1)$ and $\mathcal M(0)=\O(1)$, +\[ + \Gamma\sim e^{-1/\mathcal G(ht^{-\beta\delta})^{d-1}} +\] +with $\mathcal G(X)=\O(X)$. This establishes the form of $\im F$ +besides the prefactor. Results from field theory predict that, for small $H$ +and $10$. $\pd{}h=\pd Xh\pd{}X=t^{-\Delta}\pd{}X$. +Unfortunately this integral doesn't converge, and it seems we cannot evaluate +this result at the level of truncation we've chosen. However, + +\[ + F(H)=At^{2-\alpha}\sum_{n=0}^\infty f_nX^n +\] +\[ + f_n=\frac1\pi\int_{-\infty}^0\frac{\im F(X)}{X^{n+1}}\,\dd X + =\frac{(-1)^{n+1}}\pi\int_0^{\infty}\frac{Xe^{-1/BX}}{X^{n+1}}\,\dd X + =\frac1\pi(-1)^{n+1}B^{n-1}\Gamma(n-1) +\] +for $n>1$. + +\begin{align} + \chi + &=\pd[2]Fh + =t^{-2\Delta}\pd[2]FX + =-\frac{2}\pi At^{2-2\Delta}\int_0^\infty\frac{X'e^{-1/BX'}}{(X+X')^3}\,\dd + X'\\ + &=\frac2\pi + \frac{ABt^{-\gamma}}{(BX)^3}\big[BX(1-BX)+e^{1/BX}\ei(-1/BX)\big] +\end{align} + +\[ + \lim_{X\to0}\chi=-\frac4\pi ABt^{-\gamma} +\] + +\[ + \beta^{-1}\chi=C_{0\pm}|t|^{-7/4}+C_{1\pm}|t|^{-3/4}+\O(1) +\] +$C_{0-}=0.025\,536\,971\,9$ $C_{1-}=-0.001\,989\,410\,7$ +\cite{barouch.1973.susceptibility} + +CORRECTIONS TO SCALING, $u_t$ and $u_h$ instead of $t$ and $h$. + +\begin{align} + u_h + &=h[1+c_ht+dht^2+e_hh^2+f_ht^3+\O(t^4,th^2)]\\ + u_t + &=t+b_th^2+c_t^2+d_t^3+e_tth^2+f_tt^4+\O(t^5,t^2h^2,h^4) +\end{align} +\begin{align} + c_h=\frac{\beta_c}{\sqrt2} + && + d_h=\frac{23\beta_c^2}{16} + && + f_h=\frac{191\beta_c^3}{48\sqrt2}\\ + c_t=\frac{\beta_c}{\sqrt2} + && + d_t=\frac{7\beta_c^2}6 + && + f_t=\frac{17\beta_c^3}{6\sqrt2}\\ + e_t=b_t\beta_c\sqrt2 + && + b_t=-\frac{E_0\pi}{16\beta_c^2} +\end{align} +$E_0=0.040\,325\,5003$ $e_h=-0.007\,27(15)$ +\[ + F(t,h)-F(t,0)=\sum_{n=1}^\infty\frac1{(2n)!}\chi_{2n}(t)h^{2n} +\] +\[ + \chi(t,h)=\pd[2]Fh=\chi_2(t)+\sum_{n=1}^\infty\frac1{(2n)!}\chi_{2(n+1)}h^{2n} +\] + +\begin{align} + \chi + &=\pd[2]Fh + =\pd[2]{F_\b}h + +\frac d{y_t}\bigg(\frac d{y_t}-1\bigg)|u_t|^{d/y_t-2}\bigg(\pd{u_t}h\bigg)^2 +\end{align} + +\input{figs/scaling_func.tex} + +\bibliographystyle{plain} +\bibliography{essential_ising} + +\end{document} + diff --git a/figs/scaling_func.gplot b/figs/scaling_func.gplot new file mode 100644 index 0000000..e25b340 --- /dev/null +++ b/figs/scaling_func.gplot @@ -0,0 +1,16 @@ + +set terminal latex rotate +set ylabel "$\\chi B/At^{-\\gamma}$" +set xlabel "$h/Bt^{\\beta\\delta}$" +set key off + +gamma = 0.57721566490153286060651209008240243104215933593992 +Ei(m, x) = gamma + log(abs(x)) + sum[k = 1 : m] x**k / (k * k!) +Xi(m, A, B, x) = -A/(x * pi) * (1-B/x-(B/x)**2 * exp(B/x)* Ei(100, -B/x)) + +set yrange [0.001:0.6] +#set logscale xy + +plot [0.001:5] Xi(200, -1, 1, x) + + diff --git a/figs/scaling_func.tex b/figs/scaling_func.tex new file mode 100644 index 0000000..09448b7 --- /dev/null +++ b/figs/scaling_func.tex @@ -0,0 +1,260 @@ +% GNUPLOT: LaTeX picture +\setlength{\unitlength}{0.240900pt} +\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi +\sbox{\plotpoint}{\rule[-0.200pt]{0.400pt}{0.400pt}}% +\begin{picture}(1500,900)(0,0) +\sbox{\plotpoint}{\rule[-0.200pt]{0.400pt}{0.400pt}}% +\put(151.0,130.0){\rule[-0.200pt]{4.818pt}{0.400pt}} +\put(131,130){\makebox(0,0)[r]{$0$}} +\put(1419.0,130.0){\rule[-0.200pt]{4.818pt}{0.400pt}} +\put(151.0,251.0){\rule[-0.200pt]{4.818pt}{0.400pt}} +\put(131,251){\makebox(0,0)[r]{$0.1$}} 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+\put(151.0,859.0){\rule[-0.200pt]{310.279pt}{0.400pt}} +\end{picture} diff --git a/makefile b/makefile new file mode 100644 index 0000000..a2a0925 --- /dev/null +++ b/makefile @@ -0,0 +1,13 @@ + +DOC=essential_ising.tex +FIGS=scaling_func + +all: ${FIGS:%=figs/%.tex} + rubber --pdf $(DOC) + +figs/%.tex: figs/%.gplot + gnuplot $< > $@ + +clean: + rubber --clean $(DOC) + -- cgit v1.2.3-70-g09d2 From d16d1e6b3394f7cd604cb9c64630042c8c77d6dc Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 26 May 2017 10:43:44 -0400 Subject: moved files to new name --- essential-ising.bib | 99 +++++++++++++++++++++++++ essential-ising.tex | 206 ++++++++++++++++++++++++++++++++++++++++++++++++++++ essential_ising.bib | 99 ------------------------- essential_ising.tex | 206 ---------------------------------------------------- 4 files changed, 305 insertions(+), 305 deletions(-) create mode 100644 essential-ising.bib create mode 100644 essential-ising.tex delete mode 100644 essential_ising.bib delete mode 100644 essential_ising.tex diff --git a/essential-ising.bib b/essential-ising.bib new file mode 100644 index 0000000..c84c1a7 --- /dev/null +++ b/essential-ising.bib @@ -0,0 +1,99 @@ + +@article{houghton.1980.metastable, + title={The metastable Ising magnet in a negative field}, + author={Hougton, Anthony and Lubensky, Tom C}, + journal={Physics Letters A}, + volume={77}, + number={6}, + pages={479--480}, + year={1980}, + publisher={North-Holland} +} + +@article{langer.1967.condensation, + title={Theory of the condensation point}, + author={Langer, James S}, + journal={Annals of Physics}, + volume={41}, + number={1}, + pages={108--157}, + year={1967}, + publisher={Elsevier} +} + +@article{langer.1969.metastable, + title={Statistical theory of the decay of metastable states}, + author={Langer, JS}, + journal={Annals of Physics}, + volume={54}, + number={2}, + pages={258--275}, + year={1969}, + publisher={Elsevier} +} + +@article{gaveau.1989.analytic, + title={Metastable decay rates and analytic continuation}, + author={Gaveau, B and Schulman, LS}, + journal={Letters in Mathematical Physics}, + volume={18}, + number={3}, + pages={201--208}, + year={1989}, + publisher={Springer} +} + +@article{bogomolny.1977.dispersion, + title={Calculation of the green functions by the coupling constant dispersion relations}, + author={Bogomolny, Evgeny B}, + journal={Physics Letters B}, + volume={67}, + number={2}, + pages={193--194}, + year={1977}, + publisher={Elsevier} +} + +@article{gunther.1993.transfer-matrix, + title={Numerical transfer-matrix study of metastability in the d= 2 Ising model}, + author={G{\"u}nther, Christoph CA and Rikvold, Per Arne and Novotny, MA}, + journal={Physical review letters}, + volume={71}, + number={24}, + pages={3898}, + year={1993}, + publisher={APS} +} + +@inproceedings{widom.1981.interface, + title={Structure of the interface between fluid phases}, + author={Widom, Benjamin}, + booktitle={Faraday Symposia of the Chemical Society}, + volume={16}, + pages={7--21}, + year={1981}, + organization={Royal Society of Chemistry} +} + +@article{gunther.1980.goldstone, + title={Goldstone modes in vacuum decay and first-order phase transitions}, + author={G{\"u}nther, NJ and Wallace, DJ and Nicole, DA}, + journal={Journal of Physics A: Mathematical and General}, + volume={13}, + number={5}, + pages={1755}, + year={1980}, + publisher={IOP Publishing} +} + +@article{barouch.1973.susceptibility, + title={Zero-field susceptibility of the two-dimensional Ising model near ${T}_c$}, + author={Barouch, Eytan and McCoy, Barry M and Wu, Tai Tsun}, + journal={Physical Review Letters}, + volume={31}, + number={23}, + pages={1409}, + year={1973}, + publisher={APS} +} + diff --git a/essential-ising.tex b/essential-ising.tex new file mode 100644 index 0000000..d6a92cc --- /dev/null +++ b/essential-ising.tex @@ -0,0 +1,206 @@ +% Ising model abrupt transition. +% +% Created by Jaron Kent-Dobias on Thu Apr 20 12:50:56 EDT 2017. +% Copyright (c) 2017 Jaron Kent-Dobias. All rights reserved. +% +\documentclass[fleqn]{article} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{amsmath,amssymb,latexsym,concmath,mathtools,xifthen,mfpic} + +\mathtoolsset{showonlyrefs=true} + +\title{Essential Singularity in the Ising Abrupt Transition} +\author{Jaron Kent-Dobias} + +\date{April 20, 2017} + +\begin{document} + +\def\[{\begin{equation}} +\def\]{\end{equation}} + +\def\im{\mathop{\mathrm{Im}}\nolimits} +\def\dd{\mathrm d} +\def\O{\mathcal O} +\def\ei{\mathop{\mathrm{Ei}}\nolimits} +\def\b{\mathrm b} + +\newcommand\pd[3][]{ + \ifthenelse{\isempty{#1}} + {\def\tmp{}} + {\def\tmp{^#1}} + \frac{\partial\tmp#2}{\partial#3\tmp} +} + +\maketitle + +\begin{abstract} +\end{abstract} + +It's long been known that the decay rate $\Gamma$ of metastable states in +statistical mechanics is often related to the metastable free energy $F$ by +\cite{langer.1967.condensation,langer.1969.metastable,gaveau.1989.analytic} +\[ + \Gamma\propto\im F +\] +What exactly is meant by `metastable free energy' is important to establish, +since formally the free energy relies on the existence of an equilibrium +state. Here one can imagine either analytic continuation of the free energy +through an abrupt phase transition, or restriction of the partition function +trace to states in the vicinity of the local free energy minimum that +characterizes the metastable state. In any case, the free energy develops a +nonzero imaginary part in the metastable region. Heuristically, this can be +thought of as similar to what happens in quantum mechanics with a non-unitary +Hamiltonian: the imaginary part describes loss of probability in the system +that corresponds to decay. + +One can estimate the scaling of the decay rate of the {\sc 2d} Ising model +using ideas from nucleation theory. In this framework, the metastable state +decays when a sufficiently large domain in the stable state forms to grow +stably to fill out the whole system. The free energy of a domain of $N$ spins +causes a free energy change +\[ + \Delta F=\Sigma N^\sigma-MHN +\] +where $\Sigma$ is the surface tension and $1-\frac1d\leq\sigma<1$. This is +maximized by +\[ + N_c=\bigg(\frac{MH}{\sigma\Sigma}\bigg)^{-1/(\sigma-1)} +\] +which corresponds to a free energy change +\[ + \Delta F_c\sim\bigg(\frac\Sigma{(MH)^\sigma}\bigg)^{1/(1-\sigma)} +\] +The rate of formation is proportional to the Boltzmann factor, +\[ + \Gamma\sim e^{-\beta \Delta + F_c}=e^{-\beta(\Sigma/(MH)^\sigma)^{1/(1-\sigma)}} +\] +For domains whose boundary is minimal, $\sigma=1-\frac1d$ and this becomes +\[ + \Gamma\sim e^{-\beta(\Sigma/(MH)^\sigma)^{d-1}} +\] +Since $\Sigma\sim t^\mu\mathcal S(ht^{-\beta\delta})$ with $\mu=-\nu+\gamma+2\beta$ +\cite{widom.1981.interface} and $M\sim t^\beta\mathcal M(ht^{-\beta\delta})$ +with $\mathcal S(0)=\O(1)$ and $\mathcal M(0)=\O(1)$, +\[ + \Gamma\sim e^{-1/\mathcal G(ht^{-\beta\delta})^{d-1}} +\] +with $\mathcal G(X)=\O(X)$. This establishes the form of $\im F$ +besides the prefactor. Results from field theory predict that, for small $H$ +and $10$. $\pd{}h=\pd Xh\pd{}X=t^{-\Delta}\pd{}X$. +Unfortunately this integral doesn't converge, and it seems we cannot evaluate +this result at the level of truncation we've chosen. However, + +\[ + F(H)=At^{2-\alpha}\sum_{n=0}^\infty f_nX^n +\] +\[ + f_n=\frac1\pi\int_{-\infty}^0\frac{\im F(X)}{X^{n+1}}\,\dd X + =\frac{(-1)^{n+1}}\pi\int_0^{\infty}\frac{Xe^{-1/BX}}{X^{n+1}}\,\dd X + =\frac1\pi(-1)^{n+1}B^{n-1}\Gamma(n-1) +\] +for $n>1$. + +\begin{align} + \chi + &=\pd[2]Fh + =t^{-2\Delta}\pd[2]FX + =-\frac{2}\pi At^{2-2\Delta}\int_0^\infty\frac{X'e^{-1/BX'}}{(X+X')^3}\,\dd + X'\\ + &=\frac2\pi + \frac{ABt^{-\gamma}}{(BX)^3}\big[BX(1-BX)+e^{1/BX}\ei(-1/BX)\big] +\end{align} + +\[ + \lim_{X\to0}\chi=-\frac4\pi ABt^{-\gamma} +\] + +\[ + \beta^{-1}\chi=C_{0\pm}|t|^{-7/4}+C_{1\pm}|t|^{-3/4}+\O(1) +\] +$C_{0-}=0.025\,536\,971\,9$ $C_{1-}=-0.001\,989\,410\,7$ +\cite{barouch.1973.susceptibility} + +CORRECTIONS TO SCALING, $u_t$ and $u_h$ instead of $t$ and $h$. + +\begin{align} + u_h + &=h[1+c_ht+dht^2+e_hh^2+f_ht^3+\O(t^4,th^2)]\\ + u_t + &=t+b_th^2+c_t^2+d_t^3+e_tth^2+f_tt^4+\O(t^5,t^2h^2,h^4) +\end{align} +\begin{align} + c_h=\frac{\beta_c}{\sqrt2} + && + d_h=\frac{23\beta_c^2}{16} + && + f_h=\frac{191\beta_c^3}{48\sqrt2}\\ + c_t=\frac{\beta_c}{\sqrt2} + && + d_t=\frac{7\beta_c^2}6 + && + f_t=\frac{17\beta_c^3}{6\sqrt2}\\ + e_t=b_t\beta_c\sqrt2 + && + b_t=-\frac{E_0\pi}{16\beta_c^2} +\end{align} +$E_0=0.040\,325\,5003$ $e_h=-0.007\,27(15)$ +\[ + F(t,h)-F(t,0)=\sum_{n=1}^\infty\frac1{(2n)!}\chi_{2n}(t)h^{2n} +\] +\[ + \chi(t,h)=\pd[2]Fh=\chi_2(t)+\sum_{n=1}^\infty\frac1{(2n)!}\chi_{2(n+1)}h^{2n} +\] + +\begin{align} + \chi + &=\pd[2]Fh + =\pd[2]{F_\b}h + +\frac d{y_t}\bigg(\frac d{y_t}-1\bigg)|u_t|^{d/y_t-2}\bigg(\pd{u_t}h\bigg)^2 +\end{align} + +\input{figs/scaling_func.tex} + +\bibliographystyle{plain} +\bibliography{essential_ising} + +\end{document} + diff --git a/essential_ising.bib b/essential_ising.bib deleted file mode 100644 index c84c1a7..0000000 --- a/essential_ising.bib +++ /dev/null @@ -1,99 +0,0 @@ - -@article{houghton.1980.metastable, - title={The metastable Ising magnet in a negative field}, - author={Hougton, Anthony and Lubensky, Tom C}, - journal={Physics Letters A}, - volume={77}, - number={6}, - pages={479--480}, - year={1980}, - publisher={North-Holland} -} - -@article{langer.1967.condensation, - title={Theory of the condensation point}, - author={Langer, James S}, - journal={Annals of Physics}, - volume={41}, - number={1}, - pages={108--157}, - year={1967}, - publisher={Elsevier} -} - -@article{langer.1969.metastable, - title={Statistical theory of the decay of metastable states}, - author={Langer, JS}, - journal={Annals of Physics}, - volume={54}, - number={2}, - pages={258--275}, - year={1969}, - publisher={Elsevier} -} - -@article{gaveau.1989.analytic, - title={Metastable decay rates and analytic continuation}, - author={Gaveau, B and Schulman, LS}, - journal={Letters in Mathematical Physics}, - volume={18}, - number={3}, - pages={201--208}, - year={1989}, - publisher={Springer} -} - -@article{bogomolny.1977.dispersion, - title={Calculation of the green functions by the coupling constant dispersion relations}, - author={Bogomolny, Evgeny B}, - journal={Physics Letters B}, - volume={67}, - number={2}, - pages={193--194}, - year={1977}, - publisher={Elsevier} -} - -@article{gunther.1993.transfer-matrix, - title={Numerical transfer-matrix study of metastability in the d= 2 Ising model}, - author={G{\"u}nther, Christoph CA and Rikvold, Per Arne and Novotny, MA}, - journal={Physical review letters}, - volume={71}, - number={24}, - pages={3898}, - year={1993}, - publisher={APS} -} - -@inproceedings{widom.1981.interface, - title={Structure of the interface between fluid phases}, - author={Widom, Benjamin}, - booktitle={Faraday Symposia of the Chemical Society}, - volume={16}, - pages={7--21}, - year={1981}, - organization={Royal Society of Chemistry} -} - -@article{gunther.1980.goldstone, - title={Goldstone modes in vacuum decay and first-order phase transitions}, - author={G{\"u}nther, NJ and Wallace, DJ and Nicole, DA}, - journal={Journal of Physics A: Mathematical and General}, - volume={13}, - number={5}, - pages={1755}, - year={1980}, - publisher={IOP Publishing} -} - -@article{barouch.1973.susceptibility, - title={Zero-field susceptibility of the two-dimensional Ising model near ${T}_c$}, - author={Barouch, Eytan and McCoy, Barry M and Wu, Tai Tsun}, - journal={Physical Review Letters}, - volume={31}, - number={23}, - pages={1409}, - year={1973}, - publisher={APS} -} - diff --git a/essential_ising.tex b/essential_ising.tex deleted file mode 100644 index d6a92cc..0000000 --- a/essential_ising.tex +++ /dev/null @@ -1,206 +0,0 @@ -% Ising model abrupt transition. -% -% Created by Jaron Kent-Dobias on Thu Apr 20 12:50:56 EDT 2017. -% Copyright (c) 2017 Jaron Kent-Dobias. All rights reserved. -% -\documentclass[fleqn]{article} - -\usepackage[utf8]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{amsmath,amssymb,latexsym,concmath,mathtools,xifthen,mfpic} - -\mathtoolsset{showonlyrefs=true} - -\title{Essential Singularity in the Ising Abrupt Transition} -\author{Jaron Kent-Dobias} - -\date{April 20, 2017} - -\begin{document} - -\def\[{\begin{equation}} -\def\]{\end{equation}} - -\def\im{\mathop{\mathrm{Im}}\nolimits} -\def\dd{\mathrm d} -\def\O{\mathcal O} -\def\ei{\mathop{\mathrm{Ei}}\nolimits} -\def\b{\mathrm b} - -\newcommand\pd[3][]{ - \ifthenelse{\isempty{#1}} - {\def\tmp{}} - {\def\tmp{^#1}} - \frac{\partial\tmp#2}{\partial#3\tmp} -} - -\maketitle - -\begin{abstract} -\end{abstract} - -It's long been known that the decay rate $\Gamma$ of metastable states in -statistical mechanics is often related to the metastable free energy $F$ by -\cite{langer.1967.condensation,langer.1969.metastable,gaveau.1989.analytic} -\[ - \Gamma\propto\im F -\] -What exactly is meant by `metastable free energy' is important to establish, -since formally the free energy relies on the existence of an equilibrium -state. Here one can imagine either analytic continuation of the free energy -through an abrupt phase transition, or restriction of the partition function -trace to states in the vicinity of the local free energy minimum that -characterizes the metastable state. In any case, the free energy develops a -nonzero imaginary part in the metastable region. Heuristically, this can be -thought of as similar to what happens in quantum mechanics with a non-unitary -Hamiltonian: the imaginary part describes loss of probability in the system -that corresponds to decay. - -One can estimate the scaling of the decay rate of the {\sc 2d} Ising model -using ideas from nucleation theory. In this framework, the metastable state -decays when a sufficiently large domain in the stable state forms to grow -stably to fill out the whole system. The free energy of a domain of $N$ spins -causes a free energy change -\[ - \Delta F=\Sigma N^\sigma-MHN -\] -where $\Sigma$ is the surface tension and $1-\frac1d\leq\sigma<1$. This is -maximized by -\[ - N_c=\bigg(\frac{MH}{\sigma\Sigma}\bigg)^{-1/(\sigma-1)} -\] -which corresponds to a free energy change -\[ - \Delta F_c\sim\bigg(\frac\Sigma{(MH)^\sigma}\bigg)^{1/(1-\sigma)} -\] -The rate of formation is proportional to the Boltzmann factor, -\[ - \Gamma\sim e^{-\beta \Delta - F_c}=e^{-\beta(\Sigma/(MH)^\sigma)^{1/(1-\sigma)}} -\] -For domains whose boundary is minimal, $\sigma=1-\frac1d$ and this becomes -\[ - \Gamma\sim e^{-\beta(\Sigma/(MH)^\sigma)^{d-1}} -\] -Since $\Sigma\sim t^\mu\mathcal S(ht^{-\beta\delta})$ with $\mu=-\nu+\gamma+2\beta$ -\cite{widom.1981.interface} and $M\sim t^\beta\mathcal M(ht^{-\beta\delta})$ -with $\mathcal S(0)=\O(1)$ and $\mathcal M(0)=\O(1)$, -\[ - \Gamma\sim e^{-1/\mathcal G(ht^{-\beta\delta})^{d-1}} -\] -with $\mathcal G(X)=\O(X)$. This establishes the form of $\im F$ -besides the prefactor. Results from field theory predict that, for small $H$ -and $10$. $\pd{}h=\pd Xh\pd{}X=t^{-\Delta}\pd{}X$. -Unfortunately this integral doesn't converge, and it seems we cannot evaluate -this result at the level of truncation we've chosen. However, - -\[ - F(H)=At^{2-\alpha}\sum_{n=0}^\infty f_nX^n -\] -\[ - f_n=\frac1\pi\int_{-\infty}^0\frac{\im F(X)}{X^{n+1}}\,\dd X - =\frac{(-1)^{n+1}}\pi\int_0^{\infty}\frac{Xe^{-1/BX}}{X^{n+1}}\,\dd X - =\frac1\pi(-1)^{n+1}B^{n-1}\Gamma(n-1) -\] -for $n>1$. - -\begin{align} - \chi - &=\pd[2]Fh - =t^{-2\Delta}\pd[2]FX - =-\frac{2}\pi At^{2-2\Delta}\int_0^\infty\frac{X'e^{-1/BX'}}{(X+X')^3}\,\dd - X'\\ - &=\frac2\pi - \frac{ABt^{-\gamma}}{(BX)^3}\big[BX(1-BX)+e^{1/BX}\ei(-1/BX)\big] -\end{align} - -\[ - \lim_{X\to0}\chi=-\frac4\pi ABt^{-\gamma} -\] - -\[ - \beta^{-1}\chi=C_{0\pm}|t|^{-7/4}+C_{1\pm}|t|^{-3/4}+\O(1) -\] -$C_{0-}=0.025\,536\,971\,9$ $C_{1-}=-0.001\,989\,410\,7$ -\cite{barouch.1973.susceptibility} - -CORRECTIONS TO SCALING, $u_t$ and $u_h$ instead of $t$ and $h$. - -\begin{align} - u_h - &=h[1+c_ht+dht^2+e_hh^2+f_ht^3+\O(t^4,th^2)]\\ - u_t - &=t+b_th^2+c_t^2+d_t^3+e_tth^2+f_tt^4+\O(t^5,t^2h^2,h^4) -\end{align} -\begin{align} - c_h=\frac{\beta_c}{\sqrt2} - && - d_h=\frac{23\beta_c^2}{16} - && - f_h=\frac{191\beta_c^3}{48\sqrt2}\\ - c_t=\frac{\beta_c}{\sqrt2} - && - d_t=\frac{7\beta_c^2}6 - && - f_t=\frac{17\beta_c^3}{6\sqrt2}\\ - e_t=b_t\beta_c\sqrt2 - && - b_t=-\frac{E_0\pi}{16\beta_c^2} -\end{align} -$E_0=0.040\,325\,5003$ $e_h=-0.007\,27(15)$ -\[ - F(t,h)-F(t,0)=\sum_{n=1}^\infty\frac1{(2n)!}\chi_{2n}(t)h^{2n} -\] -\[ - \chi(t,h)=\pd[2]Fh=\chi_2(t)+\sum_{n=1}^\infty\frac1{(2n)!}\chi_{2(n+1)}h^{2n} -\] - -\begin{align} - \chi - &=\pd[2]Fh - =\pd[2]{F_\b}h - +\frac d{y_t}\bigg(\frac d{y_t}-1\bigg)|u_t|^{d/y_t-2}\bigg(\pd{u_t}h\bigg)^2 -\end{align} - -\input{figs/scaling_func.tex} - -\bibliographystyle{plain} -\bibliography{essential_ising} - -\end{document} - -- cgit v1.2.3-70-g09d2 From ec0f5b6777bf7346855a81769530dc27f6f048f5 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 26 May 2017 11:04:28 -0400 Subject: added many relevant (and some perhaps irrelevant) entries to the bibtex file, and alphabetized --- essential-ising.bib | 269 +++++++++++++++++++++++++++++++++++++++++++--------- 1 file changed, 225 insertions(+), 44 deletions(-) diff --git a/essential-ising.bib b/essential-ising.bib index c84c1a7..8482669 100644 --- a/essential-ising.bib +++ b/essential-ising.bib @@ -1,4 +1,153 @@ +@article{aharony.1983.fields, + title={Nonlinear scaling fields and corrections to scaling near criticality}, + author={Aharony, Amnon and Fisher, Michael E}, + journal={Physical Review B}, + volume={27}, + number={7}, + pages={4394}, + year={1983}, + publisher={APS} +} + +@article{barouch.1973.susceptibility, + title={Zero-field susceptibility of the two-dimensional Ising model near ${T}_c$}, + author={Barouch, Eytan and McCoy, Barry M and Wu, Tai Tsun}, + journal={Physical Review Letters}, + volume={31}, + number={23}, + pages={1409}, + year={1973}, + publisher={APS} +} + +@article{baxter.1979.series, + title={Series expansions from corner transfer matrices: the square lattice ising model}, + author={Baxter, RJ and Enting, IG}, + journal={Journal of Statistical Physics}, + volume={21}, + number={2}, + pages={103--123}, + year={1979}, + publisher={Springer} +} + +@article{baker.1980.ising, + title={Ising model phase boundary}, + author={Baker Jr, George A and Kim, Doochul}, + journal={Journal of Physics A: Mathematical and General}, + volume={13}, + number={4}, + pages={L103}, + year={1980}, + publisher={IOP Publishing} +} + +@article{bogomolny.1977.dispersion, + title={Calculation of the green functions by the coupling constant dispersion relations}, + author={Bogomolny, Evgeny B}, + journal={Physics Letters B}, + volume={67}, + number={2}, + pages={193--194}, + year={1977}, + publisher={Elsevier} +} + +@techreport{brezin.1976.perturbation, + title={Perturbation theory at large order; 2, role of the vacuum instability}, + author={Br{\'e}zin, E and Le Guillou, Jean Claude and Zinn-Justin, Jean}, + year={1976} +} + +@article{caselle.2001.critical, + title={The critical equation of state of the two-dimensional Ising model}, + author={Caselle, Michele and Hasenbusch, Martin and Pelissetto, Andrea and Vicari, Ettore}, + journal={Journal of Physics A: Mathematical and General}, + volume={34}, + number={14}, + pages={2923}, + year={2001}, + publisher={IOP Publishing} +} + +@article{fisher.1967.condensation, + title={The theory of condensation and the critical point}, + author={Fisher, Michael E}, + journal={Physics}, + volume={3}, + number={5}, + pages={255--283}, + year={1967}, + publisher={Physics Publishing Company} +} + +@article{gaveau.1989.analytic, + title={Metastable decay rates and analytic continuation}, + author={Gaveau, B and Schulman, LS}, + journal={Letters in Mathematical Physics}, + volume={18}, + number={3}, + pages={201--208}, + year={1989}, + publisher={Springer} +} + +@article{griffiths.1967.functions, + title={Thermodynamic functions for fluids and ferromagnets near the critical point}, + author={Griffiths, Robert B}, + journal={Physical Review}, + volume={158}, + number={1}, + pages={176}, + year={1967}, + publisher={APS} +} + +@article{guida.1997.3dising, + title={3D Ising model: the scaling equation of state}, + author={Guida, Riccardo and Zinn-Justin, J}, + journal={Nuclear Physics B}, + volume={489}, + number={3}, + pages={626--652}, + year={1997}, + publisher={Elsevier} +} + +@article{gunther.1980.goldstone, + title={Goldstone modes in vacuum decay and first-order phase transitions}, + author={G{\"u}nther, NJ and Wallace, DJ and Nicole, DA}, + journal={Journal of Physics A: Mathematical and General}, + volume={13}, + number={5}, + pages={1755}, + year={1980}, + publisher={IOP Publishing} +} + +@article{gunther.1993.transfer-matrix, + title={Numerical transfer-matrix study of metastability in the $d=2$ {I}sing model}, + author={G{\"u}nther, Christoph CA and Rikvold, Per Arne and Novotny, MA}, + journal={Physical review letters}, + volume={71}, + number={24}, + pages={3898}, + year={1993}, + publisher={APS} +} + +@article{harris.1984.metastability, + title={Metastability in the (1+ 1) D Ising model: a primitive droplet model calculation}, + author={Harris, CK}, + journal={Journal of Physics A: Mathematical and General}, + volume={17}, + number={8}, + pages={1767}, + year={1984}, + publisher={IOP Publishing} +} + @article{houghton.1980.metastable, title={The metastable Ising magnet in a negative field}, author={Hougton, Anthony and Lubensky, Tom C}, @@ -10,6 +159,28 @@ publisher={North-Holland} } +@article{jacucci.1983.monte, + title={Monte Carlo computation of cluster free energies in the Ising model: a test for the validity of the capillarity approximation}, + author={Jacucci, G and Perini, A and Martin, G}, + journal={Journal of Physics A: Mathematical and General}, + volume={16}, + number={2}, + pages={369}, + year={1983}, + publisher={IOP Publishing} +} + +@article{josephson.1969.equation, + title={Equation of state near the critical point}, + author={Josephson, BD}, + journal={Journal of Physics C: Solid State Physics}, + volume={2}, + number={7}, + pages={1113}, + year={1969}, + publisher={IOP Publishing} +} + @article{langer.1967.condensation, title={Theory of the condensation point}, author={Langer, James S}, @@ -32,36 +203,68 @@ publisher={Elsevier} } -@article{gaveau.1989.analytic, - title={Metastable decay rates and analytic continuation}, - author={Gaveau, B and Schulman, LS}, - journal={Letters in Mathematical Physics}, - volume={18}, - number={3}, - pages={201--208}, - year={1989}, - publisher={Springer} +@article{lipatov.1977.divergence, + title={Divergence of the perturbation theory series and the quasiclassical theory}, + author={Lipatov, LN}, + journal={Sov. Phys. JETP}, + volume={45}, + number={2}, + pages={216--223}, + year={1977} } -@article{bogomolny.1977.dispersion, - title={Calculation of the green functions by the coupling constant dispersion relations}, - author={Bogomolny, Evgeny B}, +@article{parisi.1977.asymptotic, + title={Asymptotic estimates in perturbation theory}, + author={Parisi, Giorgio}, journal={Physics Letters B}, - volume={67}, + volume={66}, number={2}, - pages={193--194}, + pages={167--169}, year={1977}, publisher={Elsevier} } -@article{gunther.1993.transfer-matrix, - title={Numerical transfer-matrix study of metastability in the d= 2 Ising model}, - author={G{\"u}nther, Christoph CA and Rikvold, Per Arne and Novotny, MA}, - journal={Physical review letters}, - volume={71}, - number={24}, - pages={3898}, - year={1993}, +@article{privman.1982.analytic, + title={Analytic properties of thermodynamic functions at first-order phase transitions}, + author={Privman, V and Schulman, LS}, + journal={Journal of Physics A: Mathematical and General}, + volume={15}, + number={5}, + pages={L231}, + year={1982}, + publisher={IOP Publishing} +} + +@article{rudnick.1976.equations, + title={Equations of state and renormalization-group recursion relations}, + author={Rudnick, Joseph and Nelson, David R}, + journal={Physical Review B}, + volume={13}, + number={5}, + pages={2208}, + year={1976}, + publisher={APS} +} + +@article{schofield.1969.parametric, + title={Parametric representation of the equation of state near a critical point}, + author={Schofield, P}, + journal={Physical Review Letters}, + volume={22}, + number={12}, + pages={606}, + year={1969}, + publisher={APS} +} + +@article{schofield.1969.correlation, + title={Correlation between critical coefficients and critical exponents}, + author={Schofield, P and Litster, JD and Ho, John T}, + journal={Physical Review Letters}, + volume={23}, + number={19}, + pages={1098}, + year={1969}, publisher={APS} } @@ -75,25 +278,3 @@ organization={Royal Society of Chemistry} } -@article{gunther.1980.goldstone, - title={Goldstone modes in vacuum decay and first-order phase transitions}, - author={G{\"u}nther, NJ and Wallace, DJ and Nicole, DA}, - journal={Journal of Physics A: Mathematical and General}, - volume={13}, - number={5}, - pages={1755}, - year={1980}, - publisher={IOP Publishing} -} - -@article{barouch.1973.susceptibility, - title={Zero-field susceptibility of the two-dimensional Ising model near ${T}_c$}, - author={Barouch, Eytan and McCoy, Barry M and Wu, Tai Tsun}, - journal={Physical Review Letters}, - volume={31}, - number={23}, - pages={1409}, - year={1973}, - publisher={APS} -} - -- cgit v1.2.3-70-g09d2 From 143e97085c2a1816ee06dbf9c5911d965002ec86 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 26 May 2017 11:07:13 -0400 Subject: propogated name change to makefile --- makefile | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/makefile b/makefile index a2a0925..f9421b5 100644 --- a/makefile +++ b/makefile @@ -1,5 +1,5 @@ -DOC=essential_ising.tex +DOC=essential-ising.tex FIGS=scaling_func all: ${FIGS:%=figs/%.tex} -- cgit v1.2.3-70-g09d2 From b53f79b38517787d40e0242c9c87fd3e47a0de58 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 26 May 2017 11:08:07 -0400 Subject: anddddd propagated name change to bibtex call --- essential-ising.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/essential-ising.tex b/essential-ising.tex index d6a92cc..0fb23a0 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -200,7 +200,7 @@ $E_0=0.040\,325\,5003$ $e_h=-0.007\,27(15)$ \input{figs/scaling_func.tex} \bibliographystyle{plain} -\bibliography{essential_ising} +\bibliography{essential-ising} \end{document} -- cgit v1.2.3-70-g09d2 From 940baf3e7eb5369064b4e8ec49a5fe2667633f5b Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 26 May 2017 11:10:12 -0400 Subject: added fisher's trigonometric parameterization paper to the bibtex file --- essential-ising.bib | 11 +++++++++++ 1 file changed, 11 insertions(+) diff --git a/essential-ising.bib b/essential-ising.bib index 8482669..bc154ea 100644 --- a/essential-ising.bib +++ b/essential-ising.bib @@ -82,6 +82,17 @@ publisher={Physics Publishing Company} } +@article{fisher.1999.trigonometric, + title={Trigonometric models for scaling behavior near criticality}, + author={Fisher, Michael E and Zinn, Shun-yong and Upton, Paul J}, + journal={Physical Review B}, + volume={59}, + number={22}, + pages={14533}, + year={1999}, + publisher={APS} +} + @article{gaveau.1989.analytic, title={Metastable decay rates and analytic continuation}, author={Gaveau, B and Schulman, LS}, -- cgit v1.2.3-70-g09d2 From f897d46d78f69457631d1a4454806f68d203bb58 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 26 May 2017 11:41:19 -0400 Subject: did some light editing --- essential-ising.tex | 48 +++++++++++++++++++++++------------------------- 1 file changed, 23 insertions(+), 25 deletions(-) diff --git a/essential-ising.tex b/essential-ising.tex index 0fb23a0..0b631e5 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -36,9 +36,6 @@ \maketitle -\begin{abstract} -\end{abstract} - It's long been known that the decay rate $\Gamma$ of metastable states in statistical mechanics is often related to the metastable free energy $F$ by \cite{langer.1967.condensation,langer.1969.metastable,gaveau.1989.analytic} @@ -82,23 +79,27 @@ For domains whose boundary is minimal, $\sigma=1-\frac1d$ and this becomes \[ \Gamma\sim e^{-\beta(\Sigma/(MH)^\sigma)^{d-1}} \] -Since $\Sigma\sim t^\mu\mathcal S(ht^{-\beta\delta})$ with $\mu=-\nu+\gamma+2\beta$ -\cite{widom.1981.interface} and $M\sim t^\beta\mathcal M(ht^{-\beta\delta})$ -with $\mathcal S(0)=\O(1)$ and $\mathcal M(0)=\O(1)$, +There are known scaling forms for the surface tension and magnetization, $\Sigma\sim t^\mu\mathcal S(ht^{-\beta\delta})$ with $\mu=-\nu+\gamma+2\beta$ +\cite{widom.1981.interface} and $M\sim t^\beta\mathcal M(ht^{-\beta\delta})$. +Since both the surface tension and magnetization have nonzero finite values at +the first-order transition $h=0$, $\mathcal S(0)=\O(1)$ and $\mathcal +M(0)=\O(1)$. It follows that \[ \Gamma\sim e^{-1/\mathcal G(ht^{-\beta\delta})^{d-1}} \] with $\mathcal G(X)=\O(X)$. This establishes the form of $\im F$ -besides the prefactor. Results from field theory predict that, for small $H$ -and $10$. $\pd{}h=\pd Xh\pd{}X=t^{-\Delta}\pd{}X$. Unfortunately this integral doesn't converge, and it seems we cannot evaluate this result at the level of truncation we've chosen. However, -- cgit v1.2.3-70-g09d2 From 6a4e2f31e40a563333812d1d17cb32efbbdfdbf7 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 29 May 2017 01:39:21 -0400 Subject: many changes, moved to revtex format and rewrote much. added free energies for all dimensions --- .gitignore | 1 + essential-ising.bib | 24 ++++- essential-ising.tex | 269 +++++++++++++++++++++++++--------------------------- 3 files changed, 153 insertions(+), 141 deletions(-) diff --git a/.gitignore b/.gitignore index 6e3b920..330fdf6 100644 --- a/.gitignore +++ b/.gitignore @@ -3,3 +3,4 @@ *.log *.bbl *.blg +*Notes.bib diff --git a/essential-ising.bib b/essential-ising.bib index bc154ea..ceb1b85 100644 --- a/essential-ising.bib +++ b/essential-ising.bib @@ -71,6 +71,17 @@ publisher={IOP Publishing} } +@article{dimitrovic.1991.finite, + title={Finite-size effects, goldstone bosons and critical exponents in the d= 3 Heisenberg model}, + author={Dimitrovi{\'c}, I and Hasenfratz, P and Nager, J and Niedermayer, Ferenc}, + journal={Nuclear Physics B}, + volume={350}, + number={3}, + pages={893--905}, + year={1991}, + publisher={Elsevier} +} + @article{fisher.1967.condensation, title={The theory of condensation and the critical point}, author={Fisher, Michael E}, @@ -205,7 +216,7 @@ @article{langer.1969.metastable, title={Statistical theory of the decay of metastable states}, - author={Langer, JS}, + author={Langer, James S}, journal={Annals of Physics}, volume={54}, number={2}, @@ -289,3 +300,14 @@ organization={Royal Society of Chemistry} } +@article{wilson.1971.renormalization, + title={Renormalization group and critical phenomena. II. Phase-space cell analysis of critical behavior}, + author={Wilson, Kenneth G}, + journal={Physical Review B}, + volume={4}, + number={9}, + pages={3184}, + year={1971}, + publisher={APS} +} + diff --git a/essential-ising.tex b/essential-ising.tex index 0b631e5..9e4d2c3 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -3,7 +3,7 @@ % Created by Jaron Kent-Dobias on Thu Apr 20 12:50:56 EDT 2017. % Copyright (c) 2017 Jaron Kent-Dobias. All rights reserved. % -\documentclass[fleqn]{article} +\documentclass[aps,prl,reprint,fleqn]{revtex4-1} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} @@ -11,21 +11,17 @@ \mathtoolsset{showonlyrefs=true} -\title{Essential Singularity in the Ising Abrupt Transition} -\author{Jaron Kent-Dobias} - -\date{April 20, 2017} - -\begin{document} - \def\[{\begin{equation}} \def\]{\end{equation}} +\def\re{\mathop{\mathrm{Re}}\nolimits} \def\im{\mathop{\mathrm{Im}}\nolimits} \def\dd{\mathrm d} \def\O{\mathcal O} +\def\o{\mathcal o} \def\ei{\mathop{\mathrm{Ei}}\nolimits} \def\b{\mathrm b} +\def\c{\mathrm c} \newcommand\pd[3][]{ \ifthenelse{\isempty{#1}} @@ -34,18 +30,58 @@ \frac{\partial\tmp#2}{\partial#3\tmp} } +\begin{document} + +\title{Essential Singularity in the Ising Abrupt Transition} +\author{Jaron Kent-Dobias} +\author{James P.~Sethna} +\affiliation{Cornell University} + +\date{April 20, 2017} + +\begin{abstract} + Test +\end{abstract} + \maketitle +The Ising model is the canonical example of a system with a continuous phase +transition, and the study of its singular properties marked the first success +of the renormalization group ({\sc rg}) method in statistical physics +\cite{wilson.1971.renormalization}. This status makes sense: it's a simple +model whose phase transition admits {\sc rg} methods in a straightforward way, +and has exact solutions in certain dimensions and for certain parameter +restrictions. However, in one respect the Ising critical point is not simply a +continuous transition: it ends the line of abrupt phase transitions at zero +field below the critical temperature. Though typically neglected in {\sc rg} +scaling analysis of the critical point, we demonstrate that there are +numerically measurable contributions to scaling due to the abrupt transition +line that cannot be accounted for by analytic changes of control or +thermodynamic variables. + +{\sc Rg} analysis predicts that the singular part of the free energy per site +$F$ as a function of reduced temperature $t=1-\frac{T_c}T$ and field $h=H/T$ in the vicinity of the critical point takes the scaling form +$F(t,h)=g_t^{2-\alpha}\mathcal F(g_hg_t^{-\Delta})$, where +$\Delta=\beta\delta$ and $g_t$, $g_h$ are analytic functions of $t$, $h$ that +transform exactly linearly under {\sc rg}. When studying the properties of the +Ising critical point, it is nearly always assumed that $\mathcal F(X)$, the +universal scaling function, is an analytic function of $X$. However, it has +long been known that there exists an essential singularity in $\mathcal F$ at +$X=0$, though its effects have long been believed to be unobservable +\cite{fisher.1967.condensation}. With careful analysis, we have found that +assuming the presence of the essential singularity is predictive of the +scaling form of e.g. the susceptibility. + +The providence of the essential singularity can be understood using the +methods of critical droplet theory for the decay of an Ising system in a +metastable state, i.e., an equilibrium Ising state for $T0$ +subjected to a small negative external field $H<0$. It's long been known that the decay rate $\Gamma$ of metastable states in statistical mechanics is often related to the metastable free energy $F$ by -\cite{langer.1967.condensation,langer.1969.metastable,gaveau.1989.analytic} -\[ - \Gamma\propto\im F -\] -What exactly is meant by `metastable free energy' is important to establish, -since formally the free energy relies on the existence of an equilibrium -state. Here one can imagine either analytic continuation of the free energy -through an abrupt phase transition, or restriction of the partition function +$\Gamma\propto\im F$ +\cite{langer.1967.condensation,langer.1969.metastable,gaveau.1989.analytic}. +`Metastable free energy' can be thought of as either an analytic continuation of the free energy +through the abrupt phase transition, or restriction of the partition function trace to states in the vicinity of the local free energy minimum that characterizes the metastable state. In any case, the free energy develops a nonzero imaginary part in the metastable region. Heuristically, this can be @@ -53,151 +89,104 @@ thought of as similar to what happens in quantum mechanics with a non-unitary Hamiltonian: the imaginary part describes loss of probability in the system that corresponds to decay. -One can estimate the scaling of the decay rate of the {\sc 2d} Ising model -using ideas from nucleation theory. In this framework, the metastable state -decays when a sufficiently large domain in the stable state forms to grow -stably to fill out the whole system. The free energy of a domain of $N$ spins -causes a free energy change -\[ - \Delta F=\Sigma N^\sigma-MHN -\] -where $\Sigma$ is the surface tension and $1-\frac1d\leq\sigma<1$. This is -maximized by -\[ - N_c=\bigg(\frac{MH}{\sigma\Sigma}\bigg)^{-1/(\sigma-1)} -\] -which corresponds to a free energy change -\[ - \Delta F_c\sim\bigg(\frac\Sigma{(MH)^\sigma}\bigg)^{1/(1-\sigma)} -\] -The rate of formation is proportional to the Boltzmann factor, -\[ - \Gamma\sim e^{-\beta \Delta - F_c}=e^{-\beta(\Sigma/(MH)^\sigma)^{1/(1-\sigma)}} -\] -For domains whose boundary is minimal, $\sigma=1-\frac1d$ and this becomes -\[ - \Gamma\sim e^{-\beta(\Sigma/(MH)^\sigma)^{d-1}} -\] -There are known scaling forms for the surface tension and magnetization, $\Sigma\sim t^\mu\mathcal S(ht^{-\beta\delta})$ with $\mu=-\nu+\gamma+2\beta$ -\cite{widom.1981.interface} and $M\sim t^\beta\mathcal M(ht^{-\beta\delta})$. -Since both the surface tension and magnetization have nonzero finite values at -the first-order transition $h=0$, $\mathcal S(0)=\O(1)$ and $\mathcal -M(0)=\O(1)$. It follows that +In critical droplet theory, the metastable state decays when a domain of the +equilibrium state forms that is sufficiently large to grow and envelop the +whole system. Assuming the free energy cost of the surface of the droplet +scales with the number of spins $N$ like $\Sigma N^\sigma$ and that of its +bulk scales like $-MHN$, the critical droplet size scales like +$N_\c\sim(MH/\Sigma)^{-1/(1-\sigma)}$ and the free energy of the critical +droplet scales like $\Delta F_\c\sim\Sigma^{1/(1-\sigma)}(HM)^{-\sigma/(1-\sigma)}$. +Assuming domains have minimal surfaces, $\sigma=1-\frac1d$ and +$\Delta F_\c\sim\Sigma^d(HM)^{-(d-1)}$. Assuming the scaling forms +$\Sigma=g_t^\mu\mathcal S(g_hg_t^{-\Delta})$ and $M=g_t^\beta\mathcal +M(g_hg_t^{-\Delta})$ and using known hyperscaling relations +\cite{widom.1981.interface}, this implies a scaling form +\begin{align} + \Delta F_c& + \sim\mathcal S^d(g_hg_t^{-\Delta})(g_hg_t^{-\Delta}\mathcal + M(g_hg_t^{-\Delta}))^{-(d-1)}\\ + &\sim\mathcal G^{-(d-1)}(g_h g_t^{-\Delta}) +\end{align} +Since both surface tension and magnetization are finite and nonzero for $H=0$ +at $T1$ this function has an essential singularity in the invariant +combination $g_hg_t^{-\Delta}$. + +This form of $\im F$ for small $h$ is known. Henceforth we will assume $h$ and +$t$ are sufficiently small that $g_t\simeq t$, $g_h\simeq h$, and all +functions of both variables can be truncated at lowest order. We make the scaling ansatz that +the imaginary part of the metastable free energy has the same singular +behavior as the real part of the equilibrium free energy, and that for small +$t$, $h$, $\im F(t,h)=t^{2-\alpha}\mathcal H(ht^{-\Delta})$ for \[ - \im F\simeq - \begin{cases} - \big(\frac - h{t^\Delta}\big)^{-(d-3)d/2}(g^*)^{-d(d-1)/4}\exp\big[-B\big(\frac - h{|t|^\Delta}\big)^{-(d-1)}(g^*)^{-(d+1)/2}\big] & d=2,4\\ - \big(\frac - h{t^\Delta}\big)^{-7/3}(g^*)^{-8/3}\exp\big[-B\big(\frac - h{t^\Delta}\big)^{-2}(g^*)^{-2}\big] - & d=3 - \end{cases} + \mathcal H(X)=A\Theta(-X)X^\zeta e^{-1/(BX)^{d-1}} \] -with $\Delta=3-\frac\epsilon2$, $g^*=2\pi^2\frac\epsilon{n+8}$ -\cite{houghton.1980.metastable,gunther.1980.goldstone}. This is consistent -with our form above. We therefore predict that +where $\Theta$ is the Heaviside function and with $\zeta=-(d-3)d/2$ for $d=2,4$ and $\zeta=-7/3$ for $d=3$ +\cite{houghton.1980.metastable,gunther.1980.goldstone}. Assuming that $F$ is +analytic in the upper complex-$h$ plane, the real part of $F$ in the +equilibrium state can be extracted from this imaginary metastable free energy +using the Kramers--Kronig relation \[ - \im F=t^{2-\alpha}\mathcal F(ht^{-\beta\delta})^{-(d-3)d/2}e^{-1/\mathcal - G(ht^{-\beta\delta})^{d-1}} + \re F(t,h)=\frac1\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{h'-h}\,\dd h' \] -In {\sc 2d} we have +In {\sc 3d} and {\sc 4d} this can be computed explicitly given our scaling +ansatz, yielding \[ - \im F=t^2\mathcal F(ht^{-\Delta})e^{-1/\mathcal G(ht^{-\Delta})} + \begin{aligned} + \mathcal F(X)&= + \frac{AB^{1/3}}{12\pi X^2}e^{-1/(BX)^2} + \bigg[\Gamma(\tfrac16)E_{7/6}((BX)^{-2})\\ + &\hspace{10em}-4BX\Gamma(\tfrac23)E_{5/3}((BX)^{-2})\bigg] + \end{aligned} + \notag \] -with $\Delta=\beta\delta=\frac{15}8$. In terms of $X=ht^{-\Delta}$, this is \[ - \im F=t^2\mathcal F(X)e^{-1/\mathcal G(X)}\simeq At^2|X|e^{-1/B|X|} + \begin{aligned} + \mathcal F(X)&= + \frac{A}{9\pi X^2}e^{1/(BX)^3} + \Big[3\Gamma(0,(BX)^{-3})\\ + &\hspace{2em}-3\Gamma(\tfrac23)\Gamma(\tfrac13,(BX)^{-3}) + -\Gamma(\tfrac13)\Gamma(-\tfrac13,(BX)^{-3})\Big] + \end{aligned} + \notag \] - -\begin{align} - F(X) - &=\frac1\pi\int_{-\infty}^\infty\frac{\im F(X')}{X'-X}\,\dd X' - =\frac{At^2}\pi\int_{-\infty}^0\frac{|X'|e^{-1/B|X'|}}{X'-X}\,\dd X'\\ - &=-\frac{At^2}\pi\int_0^\infty\frac{X'e^{-1/BX'}}{X'+X}\,\dd X' -\end{align} -since $\im F=0$ for $X>0$. $\pd{}h=\pd Xh\pd{}X=t^{-\Delta}\pd{}X$. -Unfortunately this integral doesn't converge, and it seems we cannot evaluate -this result at the level of truncation we've chosen. However, - +At the level of truncation we are working at, the Kramers--Kronig relation +does not converge in {\sc 2d}. However, the higher moments can still be +extract, e.g., the susceptibility, by taking \[ - F(H)=At^{2-\alpha}\sum_{n=0}^\infty f_nX^n + \chi\propto\pd[2]Fh + =\frac2\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h' \] +This yields \[ - f_n=\frac1\pi\int_{-\infty}^0\frac{\im F(X)}{X^{n+1}}\,\dd X - =\frac{(-1)^{n+1}}\pi\int_0^{\infty}\frac{Xe^{-1/BX}}{X^{n+1}}\,\dd X - =\frac1\pi(-1)^{n+1}B^{n-1}\Gamma(n-1) + \chi=|t|^{-\gamma}\frac{C}{2(BX)^3}\big[BX(BX-1)-e^{1/BX}\ei(-1/BX)\big] \] -for $n>1$. - -\begin{align} - \chi - &=\pd[2]Fh - =t^{-2\Delta}\pd[2]FX - =-\frac{2}\pi At^{2-2\Delta}\int_0^\infty\frac{X'e^{-1/BX'}}{(X+X')^3}\,\dd - X'\\ - &=\frac2\pi - \frac{ABt^{-\gamma}}{(BX)^3}\big[BX(1-BX)+e^{1/BX}\ei(-1/BX)\big] -\end{align} - +Scaling forms for the free energy can then be extracted by integration and +comparison with known exact results at zero field, yielding \[ - \lim_{X\to0}\chi=-\frac4\pi ABt^{-\gamma} + \mathcal M(X)=\frac{D}{BX}(BX-1)e^{1/BX}\ei(-1/BX)-D+\mathcal M(0) \] - +with $\mathcal +M(0)=\big(2(\sqrt2-1)\big)^{1/4}\big((4+3\sqrt2)\sinh^{-1}1\big)^{1/8}$, and \[ - \beta^{-1}\chi=C_{0\pm}|t|^{-7/4}+C_{1\pm}|t|^{-3/4}+\O(1) + \mathcal F(X)=\mathcal F(0)+EX(\mathcal M(0)+De^{1/BX}\ei(-1/BX)) \] -$C_{0-}=0.025\,536\,971\,9$ $C_{1-}=-0.001\,989\,410\,7$ -\cite{barouch.1973.susceptibility} -CORRECTIONS TO SCALING, $u_t$ and $u_h$ instead of $t$ and $h$. - -\begin{align} - u_h - &=h[1+c_ht+dht^2+e_hh^2+f_ht^3+\O(t^4,th^2)]\\ - u_t - &=t+b_th^2+c_t^2+d_t^3+e_tth^2+f_tt^4+\O(t^5,t^2h^2,h^4) -\end{align} -\begin{align} - c_h=\frac{\beta_c}{\sqrt2} - && - d_h=\frac{23\beta_c^2}{16} - && - f_h=\frac{191\beta_c^3}{48\sqrt2}\\ - c_t=\frac{\beta_c}{\sqrt2} - && - d_t=\frac{7\beta_c^2}6 - && - f_t=\frac{17\beta_c^3}{6\sqrt2}\\ - e_t=b_t\beta_c\sqrt2 - && - b_t=-\frac{E_0\pi}{16\beta_c^2} -\end{align} -$E_0=0.040\,325\,5003$ $e_h=-0.007\,27(15)$ -\[ - F(t,h)-F(t,0)=\sum_{n=1}^\infty\frac1{(2n)!}\chi_{2n}(t)h^{2n} -\] -\[ - \chi(t,h)=\pd[2]Fh=\chi_2(t)+\sum_{n=1}^\infty\frac1{(2n)!}\chi_{2(n+1)}h^{2n} -\] - -\begin{align} - \chi - &=\pd[2]Fh - =\pd[2]{F_\b}h - +\frac d{y_t}\bigg(\frac d{y_t}-1\bigg)|u_t|^{d/y_t-2}\bigg(\pd{u_t}h\bigg)^2 -\end{align} +How predictive are these scaling forms in the proximity of the critical point +and the abrupt transition line? We used a form of the Wolff algorithm modified +to remain efficient in the presence of an external field by incorporating the +field as another spin with coupling $|H|$ to all others +\cite{dimitrovic.1991.finite}. Data was then taken for susceptibility and +magnetization for $|t|,h\leq0.1$ -\input{figs/scaling_func.tex} -\bibliographystyle{plain} +\bibliographystyle{apsrev4-1} \bibliography{essential-ising} \end{document} -- cgit v1.2.3-70-g09d2 From d1a57e6446970c2149add56c4e017bfa991c777a Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 30 May 2017 10:11:26 -0400 Subject: a lot of writing work --- essential-ising.bib | 19 +++++++ essential-ising.tex | 161 +++++++++++++++++++++++++++++++++++++--------------- 2 files changed, 135 insertions(+), 45 deletions(-) diff --git a/essential-ising.bib b/essential-ising.bib index ceb1b85..279da4a 100644 --- a/essential-ising.bib +++ b/essential-ising.bib @@ -60,6 +60,14 @@ year={1976} } +@book{cardy.1996.scaling, + title={Scaling and renormalization in statistical physics}, + author={Cardy, John}, + volume={5}, + year={1996}, + publisher={Cambridge university press} +} + @article{caselle.2001.critical, title={The critical equation of state of the two-dimensional Ising model}, author={Caselle, Michele and Hasenbusch, Martin and Pelissetto, Andrea and Vicari, Ettore}, @@ -235,6 +243,17 @@ year={1977} } +@article{onsager.1944.crystal, + title={Crystal statistics. I. A two-dimensional model with an order-disorder transition}, + author={Onsager, Lars}, + journal={Physical Review}, + volume={65}, + number={3-4}, + pages={117}, + year={1944}, + publisher={APS} +} + @article{parisi.1977.asymptotic, title={Asymptotic estimates in perturbation theory}, author={Parisi, Giorgio}, diff --git a/essential-ising.tex b/essential-ising.tex index 9e4d2c3..ef98476 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -32,7 +32,7 @@ \begin{document} -\title{Essential Singularity in the Ising Abrupt Transition} +\title{Essential Singularities in the Ising Universal Scaling Functions} \author{Jaron Kent-Dobias} \author{James P.~Sethna} \affiliation{Cornell University} @@ -40,7 +40,7 @@ \date{April 20, 2017} \begin{abstract} - Test + This is an abstract! \end{abstract} \maketitle @@ -61,14 +61,21 @@ thermodynamic variables. {\sc Rg} analysis predicts that the singular part of the free energy per site $F$ as a function of reduced temperature $t=1-\frac{T_c}T$ and field $h=H/T$ in the vicinity of the critical point takes the scaling form -$F(t,h)=g_t^{2-\alpha}\mathcal F(g_hg_t^{-\Delta})$, where +$F(t,h)=|g_t|^{2-\alpha}\mathcal F(g_h|g_t|^{-\Delta})$ \footnote{Technically we should write $\mathcal + F_{\pm}$ to indicate that the universal scaling function takes a different + form for $t<0$ and $t>0$, but we will restrict ourselves entirely to $t<0$ + and hence $\mathcal F_-$ +for the purposes of this paper.}, where $\Delta=\beta\delta$ and $g_t$, $g_h$ are analytic functions of $t$, $h$ that -transform exactly linearly under {\sc rg}. When studying the properties of the +transform exactly linearly under {\sc rg} +\cite{cardy.1996.scaling}. When studying the properties of the Ising critical point, it is nearly always assumed that $\mathcal F(X)$, the universal scaling function, is an analytic function of $X$. However, it has long been known that there exists an essential singularity in $\mathcal F$ at $X=0$, though its effects have long been believed to be unobservable -\cite{fisher.1967.condensation}. With careful analysis, we have found that +\cite{fisher.1967.condensation}, or +simply just neglected +\cite{guida.1997.3dising,schofield.1969.parametric,schofield.1969.correlation,caselle.2001.critical,josephson.1969.equation,fisher.1999.trigonometric}. With careful analysis, we have found that assuming the presence of the essential singularity is predictive of the scaling form of e.g. the susceptibility. @@ -79,7 +86,7 @@ subjected to a small negative external field $H<0$. It's long been known that the decay rate $\Gamma$ of metastable states in statistical mechanics is often related to the metastable free energy $F$ by $\Gamma\propto\im F$ -\cite{langer.1967.condensation,langer.1969.metastable,gaveau.1989.analytic}. +\cite{langer.1967.condensation,langer.1969.metastable,gaveau.1989.analytic,privman.1982.analytic}. `Metastable free energy' can be thought of as either an analytic continuation of the free energy through the abrupt phase transition, or restriction of the partition function trace to states in the vicinity of the local free energy minimum that @@ -90,22 +97,23 @@ Hamiltonian: the imaginary part describes loss of probability in the system that corresponds to decay. In critical droplet theory, the metastable state decays when a domain of the -equilibrium state forms that is sufficiently large to grow and envelop the -whole system. Assuming the free energy cost of the surface of the droplet +equilibrium state forms whose surface-energy cost for growth is outweighed by +bulk-energy gains. Assuming the free energy cost of the surface of the droplet scales with the number of spins $N$ like $\Sigma N^\sigma$ and that of its -bulk scales like $-MHN$, the critical droplet size scales like -$N_\c\sim(MH/\Sigma)^{-1/(1-\sigma)}$ and the free energy of the critical -droplet scales like $\Delta F_\c\sim\Sigma^{1/(1-\sigma)}(HM)^{-\sigma/(1-\sigma)}$. -Assuming domains have minimal surfaces, $\sigma=1-\frac1d$ and -$\Delta F_\c\sim\Sigma^d(HM)^{-(d-1)}$. Assuming the scaling forms -$\Sigma=g_t^\mu\mathcal S(g_hg_t^{-\Delta})$ and $M=g_t^\beta\mathcal -M(g_hg_t^{-\Delta})$ and using known hyperscaling relations +bulk scales like $-M|H|N$, the critical droplet size scales like +$N_\c\sim(M|H|/\Sigma)^{-1/(1-\sigma)}$ and the free energy of the critical +droplet scales like $\Delta F_\c\sim\Sigma^{1/(1-\sigma)}(M|H|)^{-\sigma/(1-\sigma)}$. +Assuming domains have minimal surfaces, as evidenced by numeric studies +\cite{gunther.1993.transfer-matrix}, $\sigma=1-\frac1d$ and +$\Delta F_\c\sim\Sigma^d(M|H|)^{-(d-1)}$. Assuming the scaling forms +$\Sigma=|g_t|^\mu\mathcal S(g_h|g_t|^{-\Delta})$ and $M=|g_t|^\beta\mathcal +M(g_h|g_t|^{-\Delta})$ and using known hyperscaling relations \cite{widom.1981.interface}, this implies a scaling form \begin{align} \Delta F_c& - \sim\mathcal S^d(g_hg_t^{-\Delta})(g_hg_t^{-\Delta}\mathcal - M(g_hg_t^{-\Delta}))^{-(d-1)}\\ - &\sim\mathcal G^{-(d-1)}(g_h g_t^{-\Delta}) + \sim\mathcal S^d(g_h|g_t|^{-\Delta})(-g_h|g_t|^{-\Delta}\mathcal + M(g_h|g_t|^{-\Delta}))^{-(d-1)}\notag\\ + &\sim\mathcal G^{-(d-1)}(g_h|g_t|^{-\Delta}) \end{align} Since both surface tension and magnetization are finite and nonzero for $H=0$ at $T1$ this function has an essential singularity in the invariant -combination $g_hg_t^{-\Delta}$. +combination $g_h|g_t|^{-\Delta}$. This form of $\im F$ for small $h$ is known. Henceforth we will assume $h$ and $t$ are sufficiently small that $g_t\simeq t$, $g_h\simeq h$, and all functions of both variables can be truncated at lowest order. We make the scaling ansatz that the imaginary part of the metastable free energy has the same singular behavior as the real part of the equilibrium free energy, and that for small -$t$, $h$, $\im F(t,h)=t^{2-\alpha}\mathcal H(ht^{-\Delta})$ for +$t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\mathcal H(h|t|^{-\Delta})$ for \[ - \mathcal H(X)=A\Theta(-X)X^\zeta e^{-1/(BX)^{d-1}} + \mathcal H(X)=A\Theta(-X)(-X)^\zeta e^{-1/(-BX)^{d-1}} + \label{eq:im.scaling} \] where $\Theta$ is the Heaviside function and with $\zeta=-(d-3)d/2$ for $d=2,4$ and $\zeta=-7/3$ for $d=3$ \cite{houghton.1980.metastable,gunther.1980.goldstone}. Assuming that $F$ is @@ -137,25 +146,21 @@ using the Kramers--Kronig relation \] In {\sc 3d} and {\sc 4d} this can be computed explicitly given our scaling ansatz, yielding -\[ - \begin{aligned} - \mathcal F(X)&= - \frac{AB^{1/3}}{12\pi X^2}e^{-1/(BX)^2} - \bigg[\Gamma(\tfrac16)E_{7/6}((BX)^{-2})\\ - &\hspace{10em}-4BX\Gamma(\tfrac23)E_{5/3}((BX)^{-2})\bigg] - \end{aligned} +\begin{align} + \mathcal F^{\text{\sc 3d}}(X)&= + \frac{AB^{1/3}}{12\pi X^2}e^{-1/(BX)^2} + \bigg[\Gamma(\tfrac16)E_{7/6}((BX)^{-2})\\ + &\hspace{10em}-4BX\Gamma(\tfrac23)E_{5/3}((BX)^{-2})\bigg] \notag -\] -\[ - \begin{aligned} - \mathcal F(X)&= - \frac{A}{9\pi X^2}e^{1/(BX)^3} - \Big[3\Gamma(0,(BX)^{-3})\\ - &\hspace{2em}-3\Gamma(\tfrac23)\Gamma(\tfrac13,(BX)^{-3}) - -\Gamma(\tfrac13)\Gamma(-\tfrac13,(BX)^{-3})\Big] - \end{aligned} +\\ + \mathcal F^{\text{\sc 4d}}(X)&= + \frac{A}{9\pi X^2}e^{1/(BX)^3} + \Big[3\Gamma(0,(BX)^{-3})\\ + &\hspace{2em}-3\Gamma(\tfrac23)\Gamma(\tfrac13,(BX)^{-3}) + -\Gamma(\tfrac13)\Gamma(-\tfrac13,(BX)^{-3})\Big] \notag -\] +\end{align} +for {\sc 4d}. At the level of truncation we are working at, the Kramers--Kronig relation does not converge in {\sc 2d}. However, the higher moments can still be extract, e.g., the susceptibility, by taking @@ -163,27 +168,93 @@ extract, e.g., the susceptibility, by taking \chi\propto\pd[2]Fh =\frac2\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h' \] -This yields +With $\chi=|t|^{-\gamma}\mathcal Y(h|t|^{-\Delta})$, this yields \[ - \chi=|t|^{-\gamma}\frac{C}{2(BX)^3}\big[BX(BX-1)-e^{1/BX}\ei(-1/BX)\big] + \mathcal Y^{\text{\sc 2d}}(X)=\frac{C}{2(BX)^3}\big[BX(BX-1)-e^{1/BX}\ei(-1/BX)\big] + \label{eq:sus_scaling} \] +for some constant $C$. Previous work at zero field suggests that +$C=C_{0-}/T_c$, with $C_{0-}=0.025\,536\,971\,9$ +\cite{barouch.1973.susceptibility}. Scaling forms for the free energy can then be extracted by integration and comparison with known exact results at zero field, yielding \[ - \mathcal M(X)=\frac{D}{BX}(BX-1)e^{1/BX}\ei(-1/BX)-D+\mathcal M(0) + \mathcal M^{\text{\sc 2d}}(X)=\frac{D}{BX}(BX-1)e^{1/BX}\ei(-1/BX)-D+\mathcal M(0) + \label{eq:mag_scaling} \] with $\mathcal -M(0)=\big(2(\sqrt2-1)\big)^{1/4}\big((4+3\sqrt2)\sinh^{-1}1\big)^{1/8}$, and +M(0)=\big(2(\sqrt2-1)\big)^{1/4}\big((4+3\sqrt2)\sinh^{-1}1\big)^{1/8}$ +\cite{onsager.1944.crystal} and +$D$ constant, and \[ - \mathcal F(X)=\mathcal F(0)+EX(\mathcal M(0)+De^{1/BX}\ei(-1/BX)) + \mathcal F^{\text{\sc 2d}}(X)=\mathcal F(0)+EX(\mathcal M(0)+De^{1/BX}\ei(-1/BX)) \] +for $\mathcal F(0)=?$ and $E$ constant. How predictive are these scaling forms in the proximity of the critical point and the abrupt transition line? We used a form of the Wolff algorithm modified to remain efficient in the presence of an external field by incorporating the field as another spin with coupling $|H|$ to all others \cite{dimitrovic.1991.finite}. Data was then taken for susceptibility and -magnetization for $|t|,h\leq0.1$ +magnetization for $|t|,h\leq0.1$. This data is plotted in Figs.~\ref{fig:sus} +and \ref{fig:mag}, along with collapses of data onto a single universal curve +in the insets of those figures. As can be seen, there is very good agreement +between our proposed functional forms and what is measured. + +\begin{figure} + \caption{Fit of scaling form \eqref{eq:sus_scaling} to numeric data.} + \label{fig:sus} +\end{figure} + +\begin{figure} + \caption{Fit of scaling form \eqref{eq:mag_scaling} to numeric data.} + \label{fig:mag} +\end{figure} + +The most accurate components of this analysis are its predictions for the +higher-order terms in the expansion +\[ + \mathcal F(X)=-\sum_{n=0}^\infty f_n(-X)^n +\] +\cite{brezin.1976.perturbation,bogomolny.1977.dispersion,lipatov.1977.divergence,parisi.1977.asymptotic} +which can be computed directly from the integral +\[ + f_n=\frac{(-1)^{n+1}}{n!}\pd{\mathcal F}X\bigg|_{X=0} + =\frac{(-1)^{n+1}}{\pi}\int_{-\infty}^\infty\frac{\mathcal H(X)}{X^{n+1}}\;\dd X +\] +Applied to our ansatz \eqref{eq:im.scaling}, we find +\begin{align} + f_n^{\text{\textsc{2d}}} + &=\frac{AB^{n-1}\Gamma(n-1)}{\pi} + && + n \geq 2 + \label{eq:fn.2d} + \\ + f_n^{\text{\textsc{3d}}} + &=\frac{AB^{n+\frac73}\Gamma(\frac n2+\frac76)}{2\pi} + \\ + f_n^{\text{\textsc{4d}}} + &=\frac{AB^{n+2}\Gamma(\frac n3+\frac23)}{3\pi} +\end{align} +The form of \eqref{eq:fn.2d} is identical to that found in +\cite{baker.1980.ising}, with our $f_n$ relating to their $a_n$ by +$a_{n-1}=2^nf_n$. +\[ + r_n=\frac{a_n}{a_{n-1}}=\frac{2f_{n+1}}{f_{n}}=2B\frac{\Gamma(n)}{\Gamma(n-1)} +\] + +We have used results from the properties of the metastable Ising ferromagnet +and the analytic nature of the free energy to derive the universal scaling +functions for the free energy, and in \textsc{2d} the magnetization and +susceptibility, in the limit of small $t$ and $h$. Because of an essential +singularity in these functions at $h=0$---the abrupt transition line---their +form cannot be modified by analytic redefinition of control or thermodynamic +variables. These predictions match the results of simulations well. Having +demonstrated that the essential singularity in thermodynamic functions at the +abrupt singularity leads to observable effects we hope that these functional +forms will be used in conjunction with traditional perturbation methods to +better express the equation of state of the Ising model in the whole of its +parameter space. \bibliographystyle{apsrev4-1} -- cgit v1.2.3-70-g09d2 From 22690a6bebdac36fbd46b065f09e0dc8e11073c6 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 30 May 2017 11:01:48 -0400 Subject: added acknowledgments --- essential-ising.tex | 7 +++++-- 1 file changed, 5 insertions(+), 2 deletions(-) diff --git a/essential-ising.tex b/essential-ising.tex index ef98476..aeff5d0 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -3,7 +3,7 @@ % Created by Jaron Kent-Dobias on Thu Apr 20 12:50:56 EDT 2017. % Copyright (c) 2017 Jaron Kent-Dobias. All rights reserved. % -\documentclass[aps,prl,reprint,fleqn]{revtex4-1} +\documentclass[aps,prl,reprint]{revtex4-1} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} @@ -256,8 +256,11 @@ forms will be used in conjunction with traditional perturbation methods to better express the equation of state of the Ising model in the whole of its parameter space. +\begin{acknowledgments} + The authors would like to thank Tom Lubensky for a reason that Jim should + really flesh out. +\end{acknowledgments} -\bibliographystyle{apsrev4-1} \bibliography{essential-ising} \end{document} -- cgit v1.2.3-70-g09d2 From 1cae91cde5efde713b16df17d6487291d96e5b30 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 30 May 2017 11:09:52 -0400 Subject: added citation to fisher scaling variables paper --- essential-ising.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/essential-ising.tex b/essential-ising.tex index aeff5d0..cbb9866 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -68,7 +68,7 @@ $F(t,h)=|g_t|^{2-\alpha}\mathcal F(g_h|g_t|^{-\Delta})$ \footnote{Technically we for the purposes of this paper.}, where $\Delta=\beta\delta$ and $g_t$, $g_h$ are analytic functions of $t$, $h$ that transform exactly linearly under {\sc rg} -\cite{cardy.1996.scaling}. When studying the properties of the +\cite{cardy.1996.scaling,aharony.1983.fields}. When studying the properties of the Ising critical point, it is nearly always assumed that $\mathcal F(X)$, the universal scaling function, is an analytic function of $X$. However, it has long been known that there exists an essential singularity in $\mathcal F$ at -- cgit v1.2.3-70-g09d2 From 76499e55eb813b66284b16423693700e86d86ae8 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 30 May 2017 22:11:46 -0400 Subject: added draft figures --- .gitignore | 2 +- essential-ising.tex | 16 +++++++++++++--- figs/fig_mag-collapse.pdf | Bin 0 -> 156076 bytes figs/fig_sus-collapse.pdf | Bin 0 -> 159531 bytes 4 files changed, 14 insertions(+), 4 deletions(-) create mode 100644 figs/fig_mag-collapse.pdf create mode 100644 figs/fig_sus-collapse.pdf diff --git a/.gitignore b/.gitignore index 330fdf6..ee70ba3 100644 --- a/.gitignore +++ b/.gitignore @@ -1,4 +1,4 @@ -*.pdf +essential-ising.pdf *.aux *.log *.bbl diff --git a/essential-ising.tex b/essential-ising.tex index cbb9866..2d8af31 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -163,7 +163,7 @@ ansatz, yielding for {\sc 4d}. At the level of truncation we are working at, the Kramers--Kronig relation does not converge in {\sc 2d}. However, the higher moments can still be -extract, e.g., the susceptibility, by taking +extracted, e.g., the susceptibility, by taking \[ \chi\propto\pd[2]Fh =\frac2\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h' @@ -202,12 +202,22 @@ in the insets of those figures. As can be seen, there is very good agreement between our proposed functional forms and what is measured. \begin{figure} - \caption{Fit of scaling form \eqref{eq:sus_scaling} to numeric data.} + \includegraphics{figs/fig_sus-collapse} + \caption{Fit of scaling form \eqref{eq:sus_scaling} to numeric data from a + $L=1024$ square-lattice Ising model. +Different colored points show different values of $t$, which vary from +$0.01$, $0.02$, \dots, $0.1$. Solid black line shows fitted form, with +$C=0.0107\pm0.0011$ and $B=0.172\pm0.036$.} \label{fig:sus} \end{figure} \begin{figure} - \caption{Fit of scaling form \eqref{eq:mag_scaling} to numeric data.} + \includegraphics{figs/fig_mag-collapse} + \caption{Fit of scaling form \eqref{eq:mag_scaling} to numeric data from a +$L=1024$ square-lattice Ising model. Different colored points show different +values of $t$, which vary from $0.01$, $0.02$, \dots, $0.1$. Solid black lines +shows fitted form, with $\mathcal M(0)=1.21055\pm0.00031$, +$D=0.09440\pm0.00033$, and $B=0.085249\pm0.00099$.} \label{fig:mag} \end{figure} diff --git a/figs/fig_mag-collapse.pdf b/figs/fig_mag-collapse.pdf new file mode 100644 index 0000000..dd82cf5 Binary files /dev/null and b/figs/fig_mag-collapse.pdf differ diff --git a/figs/fig_sus-collapse.pdf b/figs/fig_sus-collapse.pdf new file mode 100644 index 0000000..03b561e Binary files /dev/null and b/figs/fig_sus-collapse.pdf differ -- cgit v1.2.3-70-g09d2 From 59af514d62f61356aa7d10785ec0329c9fd142c8 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 30 May 2017 22:29:26 -0400 Subject: added more entries to the bib file --- essential-ising.bib | 54 +++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 54 insertions(+) diff --git a/essential-ising.bib b/essential-ising.bib index 279da4a..d8f9116 100644 --- a/essential-ising.bib +++ b/essential-ising.bib @@ -90,6 +90,17 @@ publisher={Elsevier} } +@article{enting.1980.investigation, + title={An investigation of the high-field series expansions for the square lattice Ising model}, + author={Enting, IG and Baxter, RJ}, + journal={Journal of Physics A: Mathematical and General}, + volume={13}, + number={12}, + pages={3723}, + year={1980}, + publisher={IOP Publishing} +} + @article{fisher.1967.condensation, title={The theory of condensation and the critical point}, author={Fisher, Michael E}, @@ -211,6 +222,17 @@ publisher={IOP Publishing} } +@article{klein.1976.essential, + title={Essential singularities at first-order phase transitions}, + author={Klein, W and Wallace, DJ and Zia, RKP}, + journal={Physical Review Letters}, + volume={37}, + number={11}, + pages={639}, + year={1976}, + publisher={APS} +} + @article{langer.1967.condensation, title={Theory of the condensation point}, author={Langer, James S}, @@ -243,6 +265,28 @@ year={1977} } +@article{mccraw.1978.metastability, + title={Metastability in the two-dimensional Ising model}, + author={McCraw, RJ and Schulman, LS}, + journal={Journal of Statistical Physics}, + volume={18}, + number={3}, + pages={293--308}, + year={1978}, + publisher={Springer} +} + +@article{newman.1977.metastability, + title={Metastability and the analytic continuation of eigenvalues}, + author={Newman, CM and Schulman, LS}, + journal={Journal of Mathematical Physics}, + volume={18}, + number={1}, + pages={23--30}, + year={1977}, + publisher={AIP} +} + @article{onsager.1944.crystal, title={Crystal statistics. I. A two-dimensional model with an order-disorder transition}, author={Onsager, Lars}, @@ -265,6 +309,16 @@ publisher={Elsevier} } +@article{penrose.1987.rigorous, + title={Towards a rigorous molecular theory of metastability}, + author={Penrose, O and Lebowitz, Joel L}, + journal={Fluctuation Phenomena}, + volume={7}, + pages={293--340}, + year={1987}, + publisher={North-Holland Physics Publishing} +} + @article{privman.1982.analytic, title={Analytic properties of thermodynamic functions at first-order phase transitions}, author={Privman, V and Schulman, LS}, -- cgit v1.2.3-70-g09d2 From d83487ff2b6fca941f0c87654326bf6bfa505317 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 31 May 2017 13:23:03 -0400 Subject: new figures --- figs/fig_mag-collapse.pdf | Bin 156076 -> 158224 bytes figs/fig_sus-collapse.pdf | Bin 159531 -> 163684 bytes 2 files changed, 0 insertions(+), 0 deletions(-) diff --git a/figs/fig_mag-collapse.pdf b/figs/fig_mag-collapse.pdf index dd82cf5..f9a840c 100644 Binary files a/figs/fig_mag-collapse.pdf and b/figs/fig_mag-collapse.pdf differ diff --git a/figs/fig_sus-collapse.pdf b/figs/fig_sus-collapse.pdf index 03b561e..107d230 100644 Binary files a/figs/fig_sus-collapse.pdf and b/figs/fig_sus-collapse.pdf differ -- cgit v1.2.3-70-g09d2 From 8b2c04f7245783ab2f1788f698fabd0bc4f8067a Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 5 Jun 2017 12:15:45 -0400 Subject: changed figures to gnuplot figures --- .gitignore | 5 +- essential-ising.tex | 12 +- figs/fig-mag.gplot | 33 ++ figs/fig-mag_data-func.dat | 1001 ++++++++++++++++++++++++++++++++++++++++++++ figs/fig-mag_data-num.dat | 500 ++++++++++++++++++++++ figs/fig-sus.gplot | 32 ++ figs/fig-sus_data-func.dat | 1001 ++++++++++++++++++++++++++++++++++++++++++++ figs/fig-sus_data-num.dat | 500 ++++++++++++++++++++++ figs/fig_mag-collapse.pdf | Bin 158224 -> 0 bytes figs/fig_sus-collapse.pdf | Bin 163684 -> 0 bytes figs/scaling_func.gplot | 16 - figs/scaling_func.tex | 260 ------------ makefile | 7 +- 13 files changed, 3082 insertions(+), 285 deletions(-) create mode 100644 figs/fig-mag.gplot create mode 100644 figs/fig-mag_data-func.dat create mode 100644 figs/fig-mag_data-num.dat create mode 100644 figs/fig-sus.gplot create mode 100644 figs/fig-sus_data-func.dat create mode 100644 figs/fig-sus_data-num.dat delete mode 100644 figs/fig_mag-collapse.pdf delete mode 100644 figs/fig_sus-collapse.pdf delete mode 100644 figs/scaling_func.gplot delete mode 100644 figs/scaling_func.tex diff --git a/.gitignore b/.gitignore index ee70ba3..10c9386 100644 --- a/.gitignore +++ b/.gitignore @@ -1,6 +1,9 @@ -essential-ising.pdf +*.pdf *.aux *.log *.bbl *.blg *Notes.bib +*.dvi + +figs/*.tex diff --git a/essential-ising.tex b/essential-ising.tex index 2d8af31..b9ee8a2 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -202,22 +202,24 @@ in the insets of those figures. As can be seen, there is very good agreement between our proposed functional forms and what is measured. \begin{figure} - \includegraphics{figs/fig_sus-collapse} + \input{figs/fig-sus} + %\includegraphics{figs/fig_sus-collapse} \caption{Fit of scaling form \eqref{eq:sus_scaling} to numeric data from a $L=1024$ square-lattice Ising model. Different colored points show different values of $t$, which vary from $0.01$, $0.02$, \dots, $0.1$. Solid black line shows fitted form, with -$C=0.0107\pm0.0011$ and $B=0.172\pm0.036$.} +$C=0.0111\pm0.0023$ and $B=0.173\pm0.072$.} \label{fig:sus} \end{figure} \begin{figure} - \includegraphics{figs/fig_mag-collapse} + \input{figs/fig-mag} + %\includegraphics{figs/fig_mag-collapse} \caption{Fit of scaling form \eqref{eq:mag_scaling} to numeric data from a $L=1024$ square-lattice Ising model. Different colored points show different values of $t$, which vary from $0.01$, $0.02$, \dots, $0.1$. Solid black lines -shows fitted form, with $\mathcal M(0)=1.21055\pm0.00031$, -$D=0.09440\pm0.00033$, and $B=0.085249\pm0.00099$.} +shows fitted form, with $\mathcal M(0)=1.21039\pm0.00031$, +$D=0.09400\pm0.00035$, and $B=0.0861\pm0.0010$.} \label{fig:mag} \end{figure} diff --git a/figs/fig-mag.gplot b/figs/fig-mag.gplot new file mode 100644 index 0000000..ada3d95 --- /dev/null +++ b/figs/fig-mag.gplot @@ -0,0 +1,33 @@ + +set terminal pslatex rotate size 3.417,2.111 +set multiplot + +B = 0.0861 +A = 0.094 +M0 = 1.21039 +func = "figs/fig-mag_data-func.dat" +num = "figs/fig-mag_data-num.dat" + +set samples 10000 +set key off + +set xrange [-20:1200] +set yrange [1.18:1.75] +set ylabel offset 1,0 '$Mt^{-\beta}$' +set xlabel "$ht^{-\\beta\\delta}$" + +plot num with yerrorbars pt 0, func using (10**$1 / B):(M0 + 10**($2) * A) with linespoints pt 0 lw 3 + +set size 0.6,0.6 +set origin 0.38,0.2 +set xrange [-3:3] +set yrange [1.2:1.75] +set ylabel offset 4,0 '\footnotesize$Mt^{-\beta}$' +set xlabel offset 0,0.5 '\footnotesize$ht^{-\beta\delta}$' +set xtics format '\footnotesize$10^{%g}$' -2,2,3 +set ytics format '\footnotesize %g' + +plot num using (log10($1)):2:3 with yerrorbars pt 0, func using ($1-log10(B)):(M0 + A * 10**$2) with linespoints pt 0 lw 3 + + + diff --git a/figs/fig-mag_data-func.dat b/figs/fig-mag_data-func.dat new file mode 100644 index 0000000..29653f0 --- /dev/null +++ b/figs/fig-mag_data-func.dat @@ -0,0 +1,1001 @@ +-5. -4.698976620880227 +-4.99 -4.688976675606296 +-4.98 -4.678976738233927 +-4.97 -4.668977103542201 +-4.96 -4.658977175055251 +-4.95 -4.64897725444246 +-4.94 -4.6389776168638575 +-4.93 -4.628977705544413 +-4.92 -4.618977802145736 +-4.91 -4.608977906594962 +-4.9 -4.598978269856612 +-4.89 -4.588978384162751 +-4.88 -4.578978506414977 +-4.87 -4.568978870802035 +-4.86 -4.558979003497131 +-4.85 -4.548979144167584 +-4.84 -4.538979511480103 +-4.83 -4.528979663220596 +-4.82 -4.518979823098883 +-4.81 -4.508980093110478 +-4.8 -4.498980366682533 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1.3396628534382948 0.000020923681639429064 \ No newline at end of file diff --git a/figs/fig-sus.gplot b/figs/fig-sus.gplot new file mode 100644 index 0000000..78c73f1 --- /dev/null +++ b/figs/fig-sus.gplot @@ -0,0 +1,32 @@ + +set terminal pslatex rotate size 3.417,2.111 +set multiplot + +B = 0.173 +A = 0.0111 +func = "figs/fig-sus_data-func.dat" +num = "figs/fig-sus_data-num.dat" + +set samples 10000 +set key off + +set xrange [-20:1200] +set yrange [-0.2:13] +set ylabel offset 1,0 "$\\chi t^{\\gamma}\\times 10^3$" +set xlabel "$ht^{-\\beta\\delta}$" + +plot num using 1:(10**3 * $2):(10**3 * $3) with yerrorbars pt 0, func using (10**$1 / B):(10**(3+$2) * A) with linespoints pt 0 lw 3 + +set size 0.65,0.65 +set origin 0.3,0.3 +set xrange [-3:3] +set yrange [-4.5:-1.5] +set ylabel offset 2.5,0 '\footnotesize$\chi t^\gamma$' +set xlabel offset 0,0.5 '\footnotesize$ht^{-\beta\delta}$' +set xtics format '\footnotesize$10^{%g}$' -2,2,3 +set ytics format '\footnotesize$10^{%g}$' -4,1,-2 + +plot num using (log10($1)):(log10($2)):($3/$2) with yerrorbars pt 0, func using ($1-log10(B)):($2+log10(A)) with linespoints pt 0 lw 3 + + + diff --git a/figs/fig-sus_data-func.dat b/figs/fig-sus_data-func.dat new file mode 100644 index 0000000..972c77b --- /dev/null +++ b/figs/fig-sus_data-func.dat @@ -0,0 +1,1001 @@ +-5. 0.010094128168228536 +-4.99 0.0004986394100467353 +-4.98 -0.008441735254514486 +-4.97 0.010954806238037133 +-4.96 0.002527852467923256 +-4.95 -0.005308016370920201 +-4.94 0.0114293828525401 +-4.93 0.004018232985220573 +-4.92 -0.002858226194217986 +-4.91 -0.009219894501557 +-4.9 0.005081385417224557 +-4.89 -0.0009618799060500723 +-4.88 -0.006542816311444805 +-4.87 0.005798926687913552 +-4.86 0.0004888958038767844 +-4.85 -0.004414469974308102 +-4.84 0.006246050655940169 +-4.83 0.0015752929121863565 +-4.82 -0.0027318837493295303 +-4.81 -0.00005661662281440823 +-4.8 0.002370862728926167 +-4.79 -0.0014169159468057889 +-4.78 0.0008653476550368721 +-4.77 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-\put(151.0,859.0){\rule[-0.200pt]{310.279pt}{0.400pt}} -\end{picture} diff --git a/makefile b/makefile index f9421b5..0ec0215 100644 --- a/makefile +++ b/makefile @@ -1,9 +1,10 @@ -DOC=essential-ising.tex -FIGS=scaling_func +DOC=essential-ising +FIGS=fig-sus fig-mag all: ${FIGS:%=figs/%.tex} - rubber --pdf $(DOC) + rubber $(DOC).tex + dvipdf $(DOC).dvi figs/%.tex: figs/%.gplot gnuplot $< > $@ -- cgit v1.2.3-70-g09d2 From caf973b56c9c197ca046712dc189f368864f276c Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 5 Jun 2017 12:38:07 -0400 Subject: made some minor figure changes --- figs/fig-mag.gplot | 2 +- figs/fig-sus.gplot | 4 ++-- 2 files changed, 3 insertions(+), 3 deletions(-) diff --git a/figs/fig-mag.gplot b/figs/fig-mag.gplot index ada3d95..d1ff4aa 100644 --- a/figs/fig-mag.gplot +++ b/figs/fig-mag.gplot @@ -21,7 +21,7 @@ plot num with yerrorbars pt 0, func using (10**$1 / B):(M0 + 10**($2) * A) with set size 0.6,0.6 set origin 0.38,0.2 set xrange [-3:3] -set yrange [1.2:1.75] +set yrange [1.18:1.75] set ylabel offset 4,0 '\footnotesize$Mt^{-\beta}$' set xlabel offset 0,0.5 '\footnotesize$ht^{-\beta\delta}$' set xtics format '\footnotesize$10^{%g}$' -2,2,3 diff --git a/figs/fig-sus.gplot b/figs/fig-sus.gplot index 78c73f1..8a44651 100644 --- a/figs/fig-sus.gplot +++ b/figs/fig-sus.gplot @@ -17,8 +17,8 @@ set xlabel "$ht^{-\\beta\\delta}$" plot num using 1:(10**3 * $2):(10**3 * $3) with yerrorbars pt 0, func using (10**$1 / B):(10**(3+$2) * A) with linespoints pt 0 lw 3 -set size 0.65,0.65 -set origin 0.3,0.3 +set size 0.6,0.6 +set origin 0.38,0.37 set xrange [-3:3] set yrange [-4.5:-1.5] set ylabel offset 2.5,0 '\footnotesize$\chi t^\gamma$' -- cgit v1.2.3-70-g09d2 From e539f537677e645a6804063a7bd3eac2d2e57113 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 5 Jun 2017 21:51:35 -0400 Subject: many changes, most notably removing our section on the high-order terms and a lot of rewriting --- essential-ising.bib | 11 +++ essential-ising.tex | 238 +++++++++++++++++++++++++--------------------------- figs/fig-mag.gplot | 4 +- figs/fig-sus.gplot | 6 +- makefile | 3 + 5 files changed, 131 insertions(+), 131 deletions(-) diff --git a/essential-ising.bib b/essential-ising.bib index d8f9116..1bdee59 100644 --- a/essential-ising.bib +++ b/essential-ising.bib @@ -265,6 +265,17 @@ year={1977} } +@article{lowe.1980.instantons, + title={Instantons and the Ising model below Tc}, + author={Lowe, MJ and Wallace, DJ}, + journal={Journal of Physics A: Mathematical and General}, + volume={13}, + number={10}, + pages={L381}, + year={1980}, + publisher={IOP Publishing} +} + @article{mccraw.1978.metastability, title={Metastability in the two-dimensional Ising model}, author={McCraw, RJ and Schulman, LS}, diff --git a/essential-ising.tex b/essential-ising.tex index b9ee8a2..4bab30f 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -6,8 +6,7 @@ \documentclass[aps,prl,reprint]{revtex4-1} \usepackage[utf8]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{amsmath,amssymb,latexsym,concmath,mathtools,xifthen,mfpic} +\usepackage{amsmath,amssymb,latexsym,mathtools,xifthen} \mathtoolsset{showonlyrefs=true} @@ -17,7 +16,7 @@ \def\re{\mathop{\mathrm{Re}}\nolimits} \def\im{\mathop{\mathrm{Im}}\nolimits} \def\dd{\mathrm d} -\def\O{\mathcal O} +\def\O{O} \def\o{\mathcal o} \def\ei{\mathop{\mathrm{Ei}}\nolimits} \def\b{\mathrm b} @@ -37,64 +36,74 @@ \author{James P.~Sethna} \affiliation{Cornell University} -\date{April 20, 2017} +\date\today \begin{abstract} - This is an abstract! + Renormalization group ideas and results from critical droplet theory are + used to construct a scaling ansatz for the imaginary component of the free + energy of an Ising model in its metastable state close to the critical + point. The analytic properties of the free energy are used to determine + asymptotic scaling functions for the free energy in the vicinity of the + critical point and the abrupt transition line. These functions have + essential singularities at zero field. Analogous forms for the magnetization + and susceptibility in two-dimensions are fit to numeric data and show good + agreement. \end{abstract} \maketitle The Ising model is the canonical example of a system with a continuous phase transition, and the study of its singular properties marked the first success -of the renormalization group ({\sc rg}) method in statistical physics +of the renormalization group (\textsc{rg}) method in statistical physics \cite{wilson.1971.renormalization}. This status makes sense: it's a simple -model whose phase transition admits {\sc rg} methods in a straightforward way, +model whose phase transition admits \textsc{rg} methods in a straightforward way, and has exact solutions in certain dimensions and for certain parameter restrictions. However, in one respect the Ising critical point is not simply a continuous transition: it ends the line of abrupt phase transitions at zero -field below the critical temperature. Though typically neglected in {\sc rg} -scaling analysis of the critical point, we demonstrate that there are +field below the critical temperature. Though typically neglected in \textsc{rg} +scaling analyses of the critical point, we demonstrate that there are numerically measurable contributions to scaling due to the abrupt transition line that cannot be accounted for by analytic changes of control or thermodynamic variables. -{\sc Rg} analysis predicts that the singular part of the free energy per site -$F$ as a function of reduced temperature $t=1-\frac{T_c}T$ and field $h=H/T$ in the vicinity of the critical point takes the scaling form -$F(t,h)=|g_t|^{2-\alpha}\mathcal F(g_h|g_t|^{-\Delta})$ \footnote{Technically we should write $\mathcal - F_{\pm}$ to indicate that the universal scaling function takes a different - form for $t<0$ and $t>0$, but we will restrict ourselves entirely to $t<0$ - and hence $\mathcal F_-$ -for the purposes of this paper.}, where -$\Delta=\beta\delta$ and $g_t$, $g_h$ are analytic functions of $t$, $h$ that -transform exactly linearly under {\sc rg} -\cite{cardy.1996.scaling,aharony.1983.fields}. When studying the properties of the -Ising critical point, it is nearly always assumed that $\mathcal F(X)$, the -universal scaling function, is an analytic function of $X$. However, it has -long been known that there exists an essential singularity in $\mathcal F$ at -$X=0$, though its effects have long been believed to be unobservable -\cite{fisher.1967.condensation}, or -simply just neglected -\cite{guida.1997.3dising,schofield.1969.parametric,schofield.1969.correlation,caselle.2001.critical,josephson.1969.equation,fisher.1999.trigonometric}. With careful analysis, we have found that -assuming the presence of the essential singularity is predictive of the -scaling form of e.g. the susceptibility. +\textsc{Rg} analysis predicts that the singular part of the free energy per +site $F$ as a function of reduced temperature $t=1-T_c/T$ and field $h=H/T$ in +the vicinity of the critical point takes the scaling form +$F(t,h)=|g_t|^{2-\alpha}\mathcal F(g_h|g_t|^{-\Delta})$ \footnote{Technically +we should write $\mathcal F_{\pm}$ to indicate that the universal scaling +function takes a different form for $t<0$ and $t>0$, but we will restrict +ourselves entirely to $t<0$ and hence $\mathcal F_-$ for the purposes of this +paper.}, where $\Delta=\beta\delta$ and $g_t$, $g_h$ are analytic functions of +$t$, $h$ that transform exactly linearly under \textsc{rg} +\cite{cardy.1996.scaling,aharony.1983.fields}. When studying the properties of +the Ising critical point, it is nearly always assumed that $\mathcal F(X)$, +the universal scaling function, is an analytic function of $X$. However, it +has long been known that there exists an essential singularity in $\mathcal F$ +at $X=0$, though its effects have long been believed to be unobservable +\cite{fisher.1967.condensation}, or simply just neglected +\cite{guida.1997.3dising,schofield.1969.parametric,schofield.1969.correlation,caselle.2001.critical,josephson.1969.equation,fisher.1999.trigonometric}. +With careful analysis, we have found that assuming the presence of the +essential singularity is predictive of the scaling form of e.g. the +susceptibility. The providence of the essential singularity can be understood using the methods of critical droplet theory for the decay of an Ising system in a metastable state, i.e., an equilibrium Ising state for $T0$ -subjected to a small negative external field $H<0$. -It's long been known that the decay rate $\Gamma$ of metastable states in -statistical mechanics is often related to the metastable free energy $F$ by -$\Gamma\propto\im F$ -\cite{langer.1967.condensation,langer.1969.metastable,gaveau.1989.analytic,privman.1982.analytic}. -`Metastable free energy' can be thought of as either an analytic continuation of the free energy -through the abrupt phase transition, or restriction of the partition function -trace to states in the vicinity of the local free energy minimum that -characterizes the metastable state. In any case, the free energy develops a -nonzero imaginary part in the metastable region. Heuristically, this can be -thought of as similar to what happens in quantum mechanics with a non-unitary -Hamiltonian: the imaginary part describes loss of probability in the system -that corresponds to decay. +subjected to a small negative external field $H<0$. The existence of an +essential singularity has also been suggested by transfer matrix +\cite{mccraw.1978.metastability,enting.1980.investigation} and \textsc{rg} +methods \cite{klein.1976.essential}. It has long been known that the decay +rate $\Gamma$ of metastable states in statistical mechanics is often related +to the metastable free energy $F$ by $\Gamma\propto\im F$ +\cite{langer.1969.metastable,penrose.1987.rigorous,gaveau.1989.analytic,privman.1982.analytic}. +`Metastable free energy' can be thought of as either an analytic continuation +of the free energy through the abrupt phase transition, or restriction of the +partition function trace to states in the vicinity of the local free energy +minimum that characterizes the metastable state. In any case, the free energy +develops a nonzero imaginary part in the metastable region. Heuristically, +this can be thought of as similar to what happens in quantum mechanics with a +non-unitary Hamiltonian: the imaginary part describes loss of probability in +the system that corresponds to decay. In critical droplet theory, the metastable state decays when a domain of the equilibrium state forms whose surface-energy cost for growth is outweighed by @@ -102,10 +111,11 @@ bulk-energy gains. Assuming the free energy cost of the surface of the droplet scales with the number of spins $N$ like $\Sigma N^\sigma$ and that of its bulk scales like $-M|H|N$, the critical droplet size scales like $N_\c\sim(M|H|/\Sigma)^{-1/(1-\sigma)}$ and the free energy of the critical -droplet scales like $\Delta F_\c\sim\Sigma^{1/(1-\sigma)}(M|H|)^{-\sigma/(1-\sigma)}$. -Assuming domains have minimal surfaces, as evidenced by numeric studies -\cite{gunther.1993.transfer-matrix}, $\sigma=1-\frac1d$ and -$\Delta F_\c\sim\Sigma^d(M|H|)^{-(d-1)}$. Assuming the scaling forms +droplet scales like $\Delta +F_\c\sim\Sigma^{1/(1-\sigma)}(M|H|)^{-\sigma/(1-\sigma)}$. Assuming domains +have minimal surfaces, as evidenced by transfer matrix studies +\cite{gunther.1993.transfer-matrix}, $\sigma=1-\frac1d$ and $\Delta +F_\c\sim\Sigma^d(M|H|)^{-(d-1)}$. Assuming the singular scaling forms $\Sigma=|g_t|^\mu\mathcal S(g_h|g_t|^{-\Delta})$ and $M=|g_t|^\beta\mathcal M(g_h|g_t|^{-\Delta})$ and using known hyperscaling relations \cite{widom.1981.interface}, this implies a scaling form @@ -113,81 +123,86 @@ M(g_h|g_t|^{-\Delta})$ and using known hyperscaling relations \Delta F_c& \sim\mathcal S^d(g_h|g_t|^{-\Delta})(-g_h|g_t|^{-\Delta}\mathcal M(g_h|g_t|^{-\Delta}))^{-(d-1)}\notag\\ - &\sim\mathcal G^{-(d-1)}(g_h|g_t|^{-\Delta}) + &\sim\mathcal G^{-(d-1)}(g_h|g_t|^{-\Delta}). \end{align} Since both surface tension and magnetization are finite and nonzero for $H=0$ -at $T1$ this function has an essential singularity in the invariant combination $g_h|g_t|^{-\Delta}$. -This form of $\im F$ for small $h$ is known. Henceforth we will assume $h$ and -$t$ are sufficiently small that $g_t\simeq t$, $g_h\simeq h$, and all -functions of both variables can be truncated at lowest order. We make the scaling ansatz that -the imaginary part of the metastable free energy has the same singular -behavior as the real part of the equilibrium free energy, and that for small -$t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\mathcal H(h|t|^{-\Delta})$ for +This form of $\im F$ for small $h$ is well known +\cite{langer.1967.condensation,harris.1984.metastability}. Henceforth we will +assume $h$ and $t$ are sufficiently small that $g_t\simeq t$, $g_h\simeq h$, +and all functions of both variables can be truncated at lowest order. We make +the scaling ansatz that the imaginary part of the metastable free energy has +the same singular behavior as the real part of the equilibrium free energy, +and that for small $t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\mathcal +H(h|t|^{-\Delta})$ for \[ - \mathcal H(X)=A\Theta(-X)(-X)^\zeta e^{-1/(-BX)^{d-1}} + \mathcal H(X)=A\Theta(-X)(-X)^\zeta e^{-1/(-BX)^{d-1}}, \label{eq:im.scaling} \] -where $\Theta$ is the Heaviside function and with $\zeta=-(d-3)d/2$ for $d=2,4$ and $\zeta=-7/3$ for $d=3$ -\cite{houghton.1980.metastable,gunther.1980.goldstone}. Assuming that $F$ is -analytic in the upper complex-$h$ plane, the real part of $F$ in the -equilibrium state can be extracted from this imaginary metastable free energy -using the Kramers--Kronig relation +where $\Theta$ is the Heaviside function. Results from combining an analysis +of fluctuations on the surface of critical droplets with \textsc{rg} recursion +relations suggest that $\zeta=-(d-3)d/2$ for $d=2,4$ and $\zeta=-7/3$ for +$d=3$ +\cite{houghton.1980.metastable,rudnick.1976.equations,gunther.1980.goldstone}. +Assuming that $F$ is analytic in the upper complex-$h$ plane, the real part of +$F$ in the equilibrium state can be extracted from this imaginary metastable +free energy using the Kramers--Kronig relation \[ - \re F(t,h)=\frac1\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{h'-h}\,\dd h' + \re F(t,h)=\frac1\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{h'-h}\,\dd h'. \] -In {\sc 3d} and {\sc 4d} this can be computed explicitly given our scaling -ansatz, yielding +This relationship has been used to compute high-order moments of the free +energy in $H$ in good agreement with transfer matrix expansions +\cite{lowe.1980.instantons}. Here, we compute the integral to come to explicit +functional forms. In \textsc{3d} and \textsc{4d} this can be computed +explicitly given our scaling ansatz, yielding \begin{align} - \mathcal F^{\text{\sc 3d}}(X)&= + \mathcal F^{\text{\textsc{3d}}}(X)&= \frac{AB^{1/3}}{12\pi X^2}e^{-1/(BX)^2} \bigg[\Gamma(\tfrac16)E_{7/6}((BX)^{-2})\\ &\hspace{10em}-4BX\Gamma(\tfrac23)E_{5/3}((BX)^{-2})\bigg] \notag \\ - \mathcal F^{\text{\sc 4d}}(X)&= + \mathcal F^{\text{\textsc{4d}}}(X)&= \frac{A}{9\pi X^2}e^{1/(BX)^3} \Big[3\Gamma(0,(BX)^{-3})\\ &\hspace{2em}-3\Gamma(\tfrac23)\Gamma(\tfrac13,(BX)^{-3}) -\Gamma(\tfrac13)\Gamma(-\tfrac13,(BX)^{-3})\Big] \notag \end{align} -for {\sc 4d}. +for \textsc{4d}. At the level of truncation we are working at, the Kramers--Kronig relation -does not converge in {\sc 2d}. However, the higher moments can still be +does not converge in \textsc{2d}. However, the higher moments can still be extracted, e.g., the susceptibility, by taking \[ \chi\propto\pd[2]Fh - =\frac2\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h' + =\frac2\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. \] With $\chi=|t|^{-\gamma}\mathcal Y(h|t|^{-\Delta})$, this yields \[ - \mathcal Y^{\text{\sc 2d}}(X)=\frac{C}{2(BX)^3}\big[BX(BX-1)-e^{1/BX}\ei(-1/BX)\big] + \mathcal Y^{\text{\textsc{2d}}}(X)=\frac{C}{2(BX)^3}\big[BX(BX-1)-e^{1/BX}\ei(-1/BX)\big] \label{eq:sus_scaling} \] for some constant $C$. Previous work at zero field suggests that $C=C_{0-}/T_c$, with $C_{0-}=0.025\,536\,971\,9$ -\cite{barouch.1973.susceptibility}. -Scaling forms for the free energy can then be extracted by integration and -comparison with known exact results at zero field, yielding +\cite{barouch.1973.susceptibility}. Scaling forms for the free energy can +then be extracted by integration and comparison with known exact results at +zero field, yielding \[ - \mathcal M^{\text{\sc 2d}}(X)=\frac{D}{BX}(BX-1)e^{1/BX}\ei(-1/BX)-D+\mathcal M(0) + \mathcal M^{\text{\textsc{2d}}}(X)=\frac{D}{BX}(BX-1)e^{1/BX}\ei(-1/BX)-D+\mathcal M(0) \label{eq:mag_scaling} \] -with $\mathcal -M(0)=\big(2(\sqrt2-1)\big)^{1/4}\big((4+3\sqrt2)\sinh^{-1}1\big)^{1/8}$ -\cite{onsager.1944.crystal} and -$D$ constant, and +with $\mathcal M(0)=\big(2(\sqrt2-1)\big)^{1/4}\big((4+3\sqrt2)\sinh^{-1}1\big)^{1/8}$ +\cite{onsager.1944.crystal} and $D$ constant, and \[ - \mathcal F^{\text{\sc 2d}}(X)=\mathcal F(0)+EX(\mathcal M(0)+De^{1/BX}\ei(-1/BX)) + \mathcal F^{\text{\textsc{2d}}}(X)=\mathcal F(0)+EX(\mathcal M(0)+De^{1/BX}\ei(-1/BX)) \] for $\mathcal F(0)=?$ and $E$ constant. @@ -203,58 +218,29 @@ between our proposed functional forms and what is measured. \begin{figure} \input{figs/fig-sus} - %\includegraphics{figs/fig_sus-collapse} - \caption{Fit of scaling form \eqref{eq:sus_scaling} to numeric data from a - $L=1024$ square-lattice Ising model. -Different colored points show different values of $t$, which vary from -$0.01$, $0.02$, \dots, $0.1$. Solid black line shows fitted form, with -$C=0.0111\pm0.0023$ and $B=0.173\pm0.072$.} + \caption{ + Fit of scaling form \eqref{eq:sus_scaling} to numeric data. Data with + sampling error taken from Monte Carlo simulations of an $L=2048$ + square-lattice Ising model with $t=0.01,0.02,\ldots,0.1$ and + $h=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Solid line shows fitted form, + with $C=0.0111\pm0.0023$ and $B=0.173\pm0.072$. + } \label{fig:sus} \end{figure} \begin{figure} \input{figs/fig-mag} - %\includegraphics{figs/fig_mag-collapse} - \caption{Fit of scaling form \eqref{eq:mag_scaling} to numeric data from a -$L=1024$ square-lattice Ising model. Different colored points show different -values of $t$, which vary from $0.01$, $0.02$, \dots, $0.1$. Solid black lines -shows fitted form, with $\mathcal M(0)=1.21039\pm0.00031$, -$D=0.09400\pm0.00035$, and $B=0.0861\pm0.0010$.} + \caption{ + Fit of scaling form \eqref{eq:mag_scaling} to numeric data. Data with + sampling error taken from Monte Carlo simulations of an $L=2048$ + square-lattice Ising model with $t=0.01,0.02,\ldots,0.1$ and + $h=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Solid line shows fitted form, + with $\mathcal M(0)=1.21039\pm0.00031$, + $D=0.09400\pm0.00035$, and $B=0.0861\pm0.0010$. + } \label{fig:mag} \end{figure} -The most accurate components of this analysis are its predictions for the -higher-order terms in the expansion -\[ - \mathcal F(X)=-\sum_{n=0}^\infty f_n(-X)^n -\] -\cite{brezin.1976.perturbation,bogomolny.1977.dispersion,lipatov.1977.divergence,parisi.1977.asymptotic} -which can be computed directly from the integral -\[ - f_n=\frac{(-1)^{n+1}}{n!}\pd{\mathcal F}X\bigg|_{X=0} - =\frac{(-1)^{n+1}}{\pi}\int_{-\infty}^\infty\frac{\mathcal H(X)}{X^{n+1}}\;\dd X -\] -Applied to our ansatz \eqref{eq:im.scaling}, we find -\begin{align} - f_n^{\text{\textsc{2d}}} - &=\frac{AB^{n-1}\Gamma(n-1)}{\pi} - && - n \geq 2 - \label{eq:fn.2d} - \\ - f_n^{\text{\textsc{3d}}} - &=\frac{AB^{n+\frac73}\Gamma(\frac n2+\frac76)}{2\pi} - \\ - f_n^{\text{\textsc{4d}}} - &=\frac{AB^{n+2}\Gamma(\frac n3+\frac23)}{3\pi} -\end{align} -The form of \eqref{eq:fn.2d} is identical to that found in -\cite{baker.1980.ising}, with our $f_n$ relating to their $a_n$ by -$a_{n-1}=2^nf_n$. -\[ - r_n=\frac{a_n}{a_{n-1}}=\frac{2f_{n+1}}{f_{n}}=2B\frac{\Gamma(n)}{\Gamma(n-1)} -\] - We have used results from the properties of the metastable Ising ferromagnet and the analytic nature of the free energy to derive the universal scaling functions for the free energy, and in \textsc{2d} the magnetization and @@ -263,7 +249,7 @@ singularity in these functions at $h=0$---the abrupt transition line---their form cannot be modified by analytic redefinition of control or thermodynamic variables. These predictions match the results of simulations well. Having demonstrated that the essential singularity in thermodynamic functions at the -abrupt singularity leads to observable effects we hope that these functional +abrupt singularity leads to observable effects. we hope that these functional forms will be used in conjunction with traditional perturbation methods to better express the equation of state of the Ising model in the whole of its parameter space. diff --git a/figs/fig-mag.gplot b/figs/fig-mag.gplot index d1ff4aa..73ee9ec 100644 --- a/figs/fig-mag.gplot +++ b/figs/fig-mag.gplot @@ -14,7 +14,7 @@ set key off set xrange [-20:1200] set yrange [1.18:1.75] set ylabel offset 1,0 '$Mt^{-\beta}$' -set xlabel "$ht^{-\\beta\\delta}$" +set xlabel '$ht^{-\Delta}$' plot num with yerrorbars pt 0, func using (10**$1 / B):(M0 + 10**($2) * A) with linespoints pt 0 lw 3 @@ -23,7 +23,7 @@ set origin 0.38,0.2 set xrange [-3:3] set yrange [1.18:1.75] set ylabel offset 4,0 '\footnotesize$Mt^{-\beta}$' -set xlabel offset 0,0.5 '\footnotesize$ht^{-\beta\delta}$' +set xlabel offset 0,0.5 '\footnotesize$ht^{-\Delta}$' set xtics format '\footnotesize$10^{%g}$' -2,2,3 set ytics format '\footnotesize %g' diff --git a/figs/fig-sus.gplot b/figs/fig-sus.gplot index 8a44651..5ba371d 100644 --- a/figs/fig-sus.gplot +++ b/figs/fig-sus.gplot @@ -12,8 +12,8 @@ set key off set xrange [-20:1200] set yrange [-0.2:13] -set ylabel offset 1,0 "$\\chi t^{\\gamma}\\times 10^3$" -set xlabel "$ht^{-\\beta\\delta}$" +set ylabel offset 1,0 '$\chi t^{\gamma}\times 10^3$' +set xlabel '$ht^{-\Delta}$' plot num using 1:(10**3 * $2):(10**3 * $3) with yerrorbars pt 0, func using (10**$1 / B):(10**(3+$2) * A) with linespoints pt 0 lw 3 @@ -22,7 +22,7 @@ set origin 0.38,0.37 set xrange [-3:3] set yrange [-4.5:-1.5] set ylabel offset 2.5,0 '\footnotesize$\chi t^\gamma$' -set xlabel offset 0,0.5 '\footnotesize$ht^{-\beta\delta}$' +set xlabel offset 0,0.5 '\footnotesize$ht^{-\Delta}$' set xtics format '\footnotesize$10^{%g}$' -2,2,3 set ytics format '\footnotesize$10^{%g}$' -4,1,-2 diff --git a/makefile b/makefile index 0ec0215..7f43e44 100644 --- a/makefile +++ b/makefile @@ -11,4 +11,7 @@ figs/%.tex: figs/%.gplot clean: rubber --clean $(DOC) + rm -f $(DOC).pdf + rm -f $(DOC)Notes.bib + rm -f ${FIGS:%=figs/%.tex} -- cgit v1.2.3-70-g09d2 From 5f49227b29b368aeffb5f055ae0ec146ac5ee013 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 9 Jun 2017 12:07:04 -0400 Subject: added logic to format too-long equations differently if preprint is set --- essential-ising.tex | 20 ++++++++++++++++++++ 1 file changed, 20 insertions(+) diff --git a/essential-ising.tex b/essential-ising.tex index 4bab30f..b1c17e6 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -29,6 +29,11 @@ \frac{\partial\tmp#2}{\partial#3\tmp} } +\makeatletter +\newif\ifreprint +\@ifclasswith{revtex4-1}{reprint}{\reprinttrue}{\reprintfalse} +\makeatother + \begin{document} \title{Essential Singularities in the Ising Universal Scaling Functions} @@ -163,6 +168,7 @@ energy in $H$ in good agreement with transfer matrix expansions \cite{lowe.1980.instantons}. Here, we compute the integral to come to explicit functional forms. In \textsc{3d} and \textsc{4d} this can be computed explicitly given our scaling ansatz, yielding +\ifreprint \begin{align} \mathcal F^{\text{\textsc{3d}}}(X)&= \frac{AB^{1/3}}{12\pi X^2}e^{-1/(BX)^2} @@ -177,6 +183,20 @@ explicitly given our scaling ansatz, yielding -\Gamma(\tfrac13)\Gamma(-\tfrac13,(BX)^{-3})\Big] \notag \end{align} +\else +\begin{align} + \mathcal F^{\text{\textsc{3d}}}(X)&= + \frac{AB^{1/3}}{12\pi X^2}e^{-1/(BX)^2} + \bigg[\Gamma(\tfrac16)E_{7/6}((BX)^{-2}) + -4BX\Gamma(\tfrac23)E_{5/3}((BX)^{-2})\bigg] +\\ + \mathcal F^{\text{\textsc{4d}}}(X)&= + \frac{A}{9\pi X^2}e^{1/(BX)^3} + \Big[3\Gamma(0,(BX)^{-3}) + -3\Gamma(\tfrac23)\Gamma(\tfrac13,(BX)^{-3}) + -\Gamma(\tfrac13)\Gamma(-\tfrac13,(BX)^{-3})\Big] +\end{align} +\fi for \textsc{4d}. At the level of truncation we are working at, the Kramers--Kronig relation does not converge in \textsc{2d}. However, the higher moments can still be -- cgit v1.2.3-70-g09d2 From c1ded289ef899560dac7ee4a9da583456b653253 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Sat, 10 Jun 2017 14:30:43 -0400 Subject: more formatting changes, filled out the awknowledgements for real --- essential-ising.tex | 79 +++++++++++++++++++++++++++++++++++------------------ 1 file changed, 52 insertions(+), 27 deletions(-) diff --git a/essential-ising.tex b/essential-ising.tex index b1c17e6..312becf 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -3,12 +3,12 @@ % Created by Jaron Kent-Dobias on Thu Apr 20 12:50:56 EDT 2017. % Copyright (c) 2017 Jaron Kent-Dobias. All rights reserved. % -\documentclass[aps,prl,reprint]{revtex4-1} +\documentclass[aps,prl,preprint]{revtex4-1} \usepackage[utf8]{inputenc} \usepackage{amsmath,amssymb,latexsym,mathtools,xifthen} -\mathtoolsset{showonlyrefs=true} +%\mathtoolsset{showonlyrefs=true} \def\[{\begin{equation}} \def\]{\end{equation}} @@ -124,12 +124,26 @@ F_\c\sim\Sigma^d(M|H|)^{-(d-1)}$. Assuming the singular scaling forms $\Sigma=|g_t|^\mu\mathcal S(g_h|g_t|^{-\Delta})$ and $M=|g_t|^\beta\mathcal M(g_h|g_t|^{-\Delta})$ and using known hyperscaling relations \cite{widom.1981.interface}, this implies a scaling form -\begin{align} - \Delta F_c& +\def\eqcritformone{ \sim\mathcal S^d(g_h|g_t|^{-\Delta})(-g_h|g_t|^{-\Delta}\mathcal - M(g_h|g_t|^{-\Delta}))^{-(d-1)}\notag\\ - &\sim\mathcal G^{-(d-1)}(g_h|g_t|^{-\Delta}). -\end{align} + M(g_h|g_t|^{-\Delta}))^{-(d-1)} +} +\def\eqcritformtwo{ + \sim\mathcal G^{-(d-1)}(g_h|g_t|^{-\Delta}). +} +\ifreprint +\[ + \begin{aligned} + \Delta F_c&\eqcritformone + \\ + &\eqcritformtwo + \end{aligned} +\] +\else +\[ + \Delta F_c\eqcritformone\eqcritformtwo +\] +\fi Since both surface tension and magnetization are finite and nonzero for $H=0$ at $T Date: Mon, 12 Jun 2017 16:05:11 -0400 Subject: changed back to reprint --- essential-ising.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/essential-ising.tex b/essential-ising.tex index 312becf..1eadf68 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -3,7 +3,7 @@ % Created by Jaron Kent-Dobias on Thu Apr 20 12:50:56 EDT 2017. % Copyright (c) 2017 Jaron Kent-Dobias. All rights reserved. % -\documentclass[aps,prl,preprint]{revtex4-1} +\documentclass[aps,prl,reprint]{revtex4-1} \usepackage[utf8]{inputenc} \usepackage{amsmath,amssymb,latexsym,mathtools,xifthen} -- cgit v1.2.3-70-g09d2 From 1b990cf214d7e2ebf003db5f7c36565b36391633 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 14 Jun 2017 14:10:53 -0400 Subject: changed the way data is processed for the plots --- data/data_square-2048.dat | 500 ++++++++++++++++++++++++++++++++++++++++++++++ essential-ising.tex | 14 +- figs/fig-mag.gplot | 15 +- figs/fig-sus.gplot | 15 +- 4 files changed, 531 insertions(+), 13 deletions(-) create mode 100644 data/data_square-2048.dat diff --git a/data/data_square-2048.dat b/data/data_square-2048.dat new file mode 100644 index 0000000..f51b46f --- /dev/null +++ b/data/data_square-2048.dat @@ -0,0 +1,500 @@ +2048 2.199190000000000 0.000020529700000 -1.547747100244487 0.000050385320576 0.786016052647641 0.000061716852283 3.214943049661604 0.170812224778066 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2.189190000000000 0.100000000000000 -1.761041828478723 0.000030720231121 0.885815248329770 0.000013835209508 1.726747269932791 0.081533723124060 0.350228352137084 0.017483163455871 diff --git a/essential-ising.tex b/essential-ising.tex index 1eadf68..4238355 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -185,19 +185,19 @@ explicitly given our scaling ansatz, yielding \def\eqthreedeeone{ \mathcal F^{\text{\textsc{3d}}}(X)&= \frac{AB^{1/3}}{12\pi X^2}e^{-1/(BX)^2} - \bigg[\Gamma(\tfrac16)E_{7/6}((BX)^{-2}) + \bigg[\Gamma(\tfrac16)E_{7/6}(-(BX)^{-2}) } \def\eqthreedeetwo{ - -4BX\Gamma(\tfrac23)E_{5/3}((BX)^{-2})\bigg] + -4BX\Gamma(\tfrac23)E_{5/3}(-(BX)^{-2})\bigg] } \def\eqfourdeeone{ \mathcal F^{\text{\textsc{4d}}}(X)&= - \frac{A}{9\pi X^2}e^{1/(BX)^3} - \Big[3\Gamma(0,(BX)^{-3}) + -\frac{A}{9\pi X^2}e^{1/(BX)^3} + \Big[3\ei(-(BX)^{-3}) } \def\eqfourdeetwo{ - -3\Gamma(\tfrac23)\Gamma(\tfrac13,(BX)^{-3}) - -\Gamma(\tfrac13)\Gamma(-\tfrac13,(BX)^{-3})\Big] + +3\Gamma(\tfrac23)\Gamma(\tfrac13,(BX)^{-3}) + +\Gamma(\tfrac13)\Gamma(-\tfrac13,(BX)^{-3})\Big] } \ifreprint \begin{align} @@ -240,7 +240,7 @@ $C=C_{0-}/T_c$, with $C_{0-}=0.025\,536\,971\,9$ then be extracted by integration and comparison with known exact results at zero field, yielding \[ - \mathcal M^{\text{\textsc{2d}}}(X)=\frac{D}{BX}(BX-1)e^{1/BX}\ei(-1/BX)-D+\mathcal M(0) + \mathcal M^{\text{\textsc{2d}}}(X)=\mathcal M(0)+D-\frac{D}{BX}(BX-1)e^{1/BX}\ei(-1/BX) \label{eq:mag_scaling} \] with $\mathcal M(0)=\big(2(\sqrt2-1)\big)^{1/4}\big((4+3\sqrt2)\sinh^{-1}1\big)^{1/8}$ diff --git a/figs/fig-mag.gplot b/figs/fig-mag.gplot index 73ee9ec..df67fa9 100644 --- a/figs/fig-mag.gplot +++ b/figs/fig-mag.gplot @@ -2,11 +2,20 @@ set terminal pslatex rotate size 3.417,2.111 set multiplot +Tc = 2 / log(1 + sqrt(2)) +Delta = 15. / 8 +beta = 1. / 8 + +t(T) = abs((T - Tc) / T) +h(T, H) = H / T +X(T, H) = h(T, H) * t(T)**(-Delta) + B = 0.0861 A = 0.094 M0 = 1.21039 + func = "figs/fig-mag_data-func.dat" -num = "figs/fig-mag_data-num.dat" +num = "data/data_square-2048.dat" set samples 10000 set key off @@ -16,7 +25,7 @@ set yrange [1.18:1.75] set ylabel offset 1,0 '$Mt^{-\beta}$' set xlabel '$ht^{-\Delta}$' -plot num with yerrorbars pt 0, func using (10**$1 / B):(M0 + 10**($2) * A) with linespoints pt 0 lw 3 +plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, func using (10**$1 / B):(M0 + 10**($2) * A) with linespoints pt 0 lw 2 lc rgb "black" set size 0.6,0.6 set origin 0.38,0.2 @@ -27,7 +36,7 @@ set xlabel offset 0,0.5 '\footnotesize$ht^{-\Delta}$' set xtics format '\footnotesize$10^{%g}$' -2,2,3 set ytics format '\footnotesize %g' -plot num using (log10($1)):2:3 with yerrorbars pt 0, func using ($1-log10(B)):(M0 + A * 10**$2) with linespoints pt 0 lw 3 +plot num using (log10(X($2, $3))):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, func using ($1-log10(B)):(M0 + A * 10**$2) with linespoints pt 0 lw 2 lc black diff --git a/figs/fig-sus.gplot b/figs/fig-sus.gplot index 5ba371d..c56bafd 100644 --- a/figs/fig-sus.gplot +++ b/figs/fig-sus.gplot @@ -2,10 +2,19 @@ set terminal pslatex rotate size 3.417,2.111 set multiplot +Tc = 2 / log(1 + sqrt(2)) +Delta = 15. / 8 +gamma = 7. / 4 + +t(T) = abs((T - Tc) / T) +h(T, H) = H / T +X(T, H) = h(T, H) * t(T)**(-Delta) + B = 0.173 A = 0.0111 + func = "figs/fig-sus_data-func.dat" -num = "figs/fig-sus_data-num.dat" +num = "data/data_square-2048.dat" set samples 10000 set key off @@ -15,7 +24,7 @@ set yrange [-0.2:13] set ylabel offset 1,0 '$\chi t^{\gamma}\times 10^3$' set xlabel '$ht^{-\Delta}$' -plot num using 1:(10**3 * $2):(10**3 * $3) with yerrorbars pt 0, func using (10**$1 / B):(10**(3+$2) * A) with linespoints pt 0 lw 3 +plot num using (X($2, $3)):(10**3 * $10 * t($2)**gamma):(10**3 * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, func using (10**$1 / B):(10**(3+$2) * A) with linespoints pt 0 lw 2 lc rgb "black" set size 0.6,0.6 set origin 0.38,0.37 @@ -26,7 +35,7 @@ set xlabel offset 0,0.5 '\footnotesize$ht^{-\Delta}$' set xtics format '\footnotesize$10^{%g}$' -2,2,3 set ytics format '\footnotesize$10^{%g}$' -4,1,-2 -plot num using (log10($1)):(log10($2)):($3/$2) with yerrorbars pt 0, func using ($1-log10(B)):($2+log10(A)) with linespoints pt 0 lw 3 +plot num using (log10(X($2, $3))):(log10($10 * t($2)**gamma)):($11 / $10):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, func using ($1-log10(B)):($2+log10(A)) with linespoints pt 0 lw 2 lc rgb "black" -- cgit v1.2.3-70-g09d2 From a63d5e9bc9fcb50de263f478e64914bf2d2cf3c1 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 28 Jun 2017 15:19:44 -0400 Subject: added new data files --- data/data_hex-2048.dat | 315 +++++++++++++++++++++++++++++ data/data_square-1024.dat | 484 ++++++++++++++++++++++++++++++++++++++++++++ data/data_square-4096.dat | 143 +++++++++++++ data/data_tri-2048.dat | 500 ++++++++++++++++++++++++++++++++++++++++++++++ data/data_tri-4096.dat | 10 + 5 files changed, 1452 insertions(+) create mode 100644 data/data_hex-2048.dat create mode 100644 data/data_square-1024.dat create mode 100644 data/data_square-4096.dat create mode 100644 data/data_tri-2048.dat create mode 100644 data/data_tri-4096.dat diff --git a/data/data_hex-2048.dat b/data/data_hex-2048.dat new file mode 100644 index 0000000..6d23f39 --- /dev/null +++ b/data/data_hex-2048.dat @@ -0,0 +1,315 @@ +2048 1.498650000000000 0.000020529700000 -1.199656576876638 0.000025939862638 0.727356424448600 0.000091127859229 2.760762205280365 0.152851753053995 34.071871788289037 1.700499691691746 +2048 1.428650000000000 0.000020529700000 -1.289936871303250 0.000018611666470 0.864696989830902 0.000024151387548 1.456289101468041 0.072194128424336 2.452230280896828 0.122487611136236 +2048 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0.582530116072684 0.000176272028500 13.408221406686287 1.052033178592329 159.937545504615287 7.994403482194035 +4096 3.630960000000000 0.000097656300000 -2.033628133182782 0.000046745788212 0.589499784683693 0.000143160650078 12.358473292316267 1.047118455378920 115.911672612661661 5.788437397329064 -- cgit v1.2.3-70-g09d2 From 91f8ae4637f1d69241c4106b97ddb0dfe64d17ca Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 28 Jun 2017 21:37:42 -0400 Subject: more new data --- data/data_hex-2048.dat | 185 ++++++++++++++++++++++++++++++++++++++++++++++ data/data_square-4096.dat | 15 ++++ data/data_tri-4096.dat | 2 + 3 files changed, 202 insertions(+) diff --git a/data/data_hex-2048.dat b/data/data_hex-2048.dat index 6d23f39..4495cb4 100644 --- a/data/data_hex-2048.dat +++ b/data/data_hex-2048.dat @@ -313,3 +313,188 @@ 2048 1.488650000000000 0.003716270000000 -1.231035722560908 0.000021104433051 0.796590867635126 0.000035889434543 2.158455648516894 0.101872296847676 6.242074984883820 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v1.2.3-70-g09d2 From 1eb6a42b57d7c59ecc1e0b6c35db6046db0e230c Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Thu, 29 Jun 2017 19:56:03 -0400 Subject: new data --- data/data_hex-4096.dat | 3 +++ data/data_square-4096.dat | 38 ++++++++++++++++++++++++++++++++++++++ data/data_tri-4096.dat | 18 ++++++++++++++++++ 3 files changed, 59 insertions(+) create mode 100644 data/data_hex-4096.dat diff --git a/data/data_hex-4096.dat b/data/data_hex-4096.dat new file mode 100644 index 0000000..9e12fd0 --- /dev/null +++ b/data/data_hex-4096.dat @@ -0,0 +1,3 @@ +4096 1.508650000000000 0.000034526700000 -1.180865009797521 0.000013269099039 0.671396929488099 0.000067843545978 4.538660539661147 0.383991449276174 118.648406185195668 5.925577811877741 +4096 1.508650000000000 0.000029033400000 -1.180785213170988 0.000012635842448 0.670757799862285 0.000066233588472 4.421189707392974 0.353972315088198 121.475114364789249 6.066147469536134 +4096 1.508650000000000 0.000097656300000 -1.181598272328014 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0.600684148237861 +4096 3.630960000000000 0.002209710000000 -2.056847926956689 0.000042389240489 0.656786717872275 0.000057403313388 9.165977251152265 0.584839256967298 16.808982574900430 0.840173016867940 -- cgit v1.2.3-70-g09d2 From 35b3f17eb7457adde87787a127bebc7c5ba286c1 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Thu, 29 Jun 2017 19:56:38 -0400 Subject: even more data --- data/data_square-4096.dat | 7 +++++++ 1 file changed, 7 insertions(+) diff --git a/data/data_square-4096.dat b/data/data_square-4096.dat index 03383f4..62cf3d6 100644 --- a/data/data_square-4096.dat +++ b/data/data_square-4096.dat @@ -194,3 +194,10 @@ 4096 2.229190000000000 0.050000000000000 -1.636066960527202 0.000020484755998 0.842005032884039 0.000012536461105 2.321249051969551 0.113950908809398 0.869380838634294 0.043463358438902 4096 2.229190000000000 0.029730200000000 -1.592202624587075 0.000020808355681 0.821671924303589 0.000014492140014 2.483152010940521 0.124776628381649 1.204461225612763 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-1.518005012175036 0.000024312382832 0.755184553369714 0.000038824302561 3.601766068573585 0.170246443106235 9.184759239721807 0.458793007682197 +4096 2.219190000000000 0.000034526700000 -1.517995779497446 0.000025375238907 0.755204916481050 0.000039019958350 3.616883351967319 0.218791155966277 8.552396476710056 0.427344236215242 +4096 2.229190000000000 0.100000000000000 -1.728187055219790 0.000016019708494 0.871639871648398 0.000007763364001 1.800105423098186 0.092493607998351 0.422754871640377 0.021112393424439 -- cgit v1.2.3-70-g09d2 From 19b9e75413387000fb5628fafa1d031726c01fca Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 30 Jun 2017 11:43:27 -0400 Subject: more data --- data/data_hex-4096.dat | 7 +++++++ data/data_square-4096.dat | 30 ++++++++++++++++++++++++++++++ data/data_tri-4096.dat | 11 +++++++++++ 3 files changed, 48 insertions(+) diff --git a/data/data_hex-4096.dat b/data/data_hex-4096.dat index 9e12fd0..cbe3572 100644 --- a/data/data_hex-4096.dat +++ b/data/data_hex-4096.dat @@ -1,3 +1,10 @@ 4096 1.508650000000000 0.000034526700000 -1.180865009797521 0.000013269099039 0.671396929488099 0.000067843545978 4.538660539661147 0.383991449276174 118.648406185195668 5.925577811877741 4096 1.508650000000000 0.000029033400000 -1.180785213170988 0.000012635842448 0.670757799862285 0.000066233588472 4.421189707392974 0.353972315088198 121.475114364789249 6.066147469536134 4096 1.508650000000000 0.000097656300000 -1.181598272328014 0.000012162747626 0.677069769867163 0.000057949338690 3.994325674615690 0.328288101444781 90.672804720331115 4.528096987526773 +4096 1.508650000000000 0.000024414100000 -1.180721234943708 0.000011996758257 0.670281098958795 0.000064191246194 4.148529773047213 0.360715119940274 118.773077672849737 5.935376774478733 +4096 1.508650000000000 0.000082118800000 -1.181424330858719 0.000012401533450 0.675666387485863 0.000058362294799 4.248481861294092 0.330397223008703 94.090918980246727 4.701741369620178 +4096 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0.005255600000000 -1.601049329198018 0.000021886166089 0.829890009262833 0.000017814652191 2.789694587344381 0.135678062470573 1.848297916287730 0.092287309401197 +4096 2.169190000000000 0.008838830000000 -1.609842546632977 0.000019483074181 0.835512564534147 0.000015369736281 2.304660334159388 0.115311293617075 1.434249775662682 0.071663660755519 +4096 2.169190000000000 0.007432540000000 -1.606499465025119 0.000019734762905 0.833480252176712 0.000015147251517 2.379650735790914 0.121674766176101 1.401900101695781 0.070013641540243 diff --git a/essential-ising.bib b/essential-ising.bib index 1bdee59..3e39487 100644 --- a/essential-ising.bib +++ b/essential-ising.bib @@ -43,6 +43,17 @@ publisher={IOP Publishing} } +@article{betts.1971.lattice, + title={Lattice-lattice scaling and the generalized law of corresponding states}, + author={Betts, DD and Guttmann, AJ and Joyce, GS}, + journal={Journal of Physics C: Solid State Physics}, + volume={4}, + number={14}, + pages={1994}, + year={1971}, + publisher={IOP Publishing} +} + @article{bogomolny.1977.dispersion, title={Calculation of the green functions by the coupling constant dispersion relations}, author={Bogomolny, Evgeny B}, diff --git a/essential-ising.tex b/essential-ising.tex index 4238355..3241954 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -1,27 +1,54 @@ -% Ising model abrupt transition. + % % Created by Jaron Kent-Dobias on Thu Apr 20 12:50:56 EDT 2017. % Copyright (c) 2017 Jaron Kent-Dobias. All rights reserved. % -\documentclass[aps,prl,reprint]{revtex4-1} +\documentclass[aps,prl,preprint]{revtex4-1} \usepackage[utf8]{inputenc} \usepackage{amsmath,amssymb,latexsym,mathtools,xifthen} +% uncomment to label only equations that are referenced in the text %\mathtoolsset{showonlyrefs=true} +% I want labels but don't want to type out ``equation'' \def\[{\begin{equation}} \def\]{\end{equation}} +% math not built-in +\def\arcsinh{\mathop{\mathrm{arcsinh}}\nolimits} +\def\arccosh{\mathop{\mathrm{arccosh}}\nolimits} +\def\ei{\mathop{\mathrm{Ei}}\nolimits} % exponential integral Ei \def\re{\mathop{\mathrm{Re}}\nolimits} \def\im{\mathop{\mathrm{Im}}\nolimits} -\def\dd{\mathrm d} -\def\O{O} -\def\o{\mathcal o} -\def\ei{\mathop{\mathrm{Ei}}\nolimits} -\def\b{\mathrm b} +\def\sgn{\mathop{\mathrm{sgn}}\nolimits} +\def\dd{d} % differential +\def\O{O} % big O +\def\o{o} % little O + +% subscript for ``critical'' values, e.g., T_\c \def\c{\mathrm c} +% scaling functions +\def\fM{\mathcal M} % magnetization +\def\fX{\mathcal Y} % susceptibility +\def\fF{\mathcal F} % free energy +\def\fiF{\mathcal H} % imaginary free energy +\def\fS{\mathcal S} % surface tension +\def\fG{\mathcal G} % exponential factor + +% lattice types +\def\sq{\mathrm{sq}} +\def\tri{\mathrm{tri}} +\def\hex{\mathrm{hex}} + +% dimensions +\def\dim{d} +\def\twodee{\textsc{2d} } +\def\threedee{\textsc{3d} } +\def\fourdee{\textsc{4d} } + +% fancy partial derivative \newcommand\pd[3][]{ \ifthenelse{\isempty{#1}} {\def\tmp{}} @@ -29,6 +56,7 @@ \frac{\partial\tmp#2}{\partial#3\tmp} } +% used to reformat display math to fit in two-column ``reprint'' mode \makeatletter \newif\ifreprint \@ifclasswith{revtex4-1}{reprint}{\reprinttrue}{\reprintfalse} @@ -72,18 +100,18 @@ line that cannot be accounted for by analytic changes of control or thermodynamic variables. \textsc{Rg} analysis predicts that the singular part of the free energy per -site $F$ as a function of reduced temperature $t=1-T_c/T$ and field $h=H/T$ in +site $F$ as a function of reduced temperature $t=1-T_\c/T$ and field $h=H/T$ in the vicinity of the critical point takes the scaling form -$F(t,h)=|g_t|^{2-\alpha}\mathcal F(g_h|g_t|^{-\Delta})$ \footnote{Technically -we should write $\mathcal F_{\pm}$ to indicate that the universal scaling +$F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\Delta})$ \footnote{Technically +we should write $\fF_{\pm}$ to indicate that the universal scaling function takes a different form for $t<0$ and $t>0$, but we will restrict -ourselves entirely to $t<0$ and hence $\mathcal F_-$ for the purposes of this -paper.}, where $\Delta=\beta\delta$ and $g_t$, $g_h$ are analytic functions of +ourselves entirely to $t<0$ and hence $\fF_-$ for the purposes of this +paper.}, where $\Delta=\beta\delta$ and $t$, $h$ are analytic functions of $t$, $h$ that transform exactly linearly under \textsc{rg} \cite{cardy.1996.scaling,aharony.1983.fields}. When studying the properties of -the Ising critical point, it is nearly always assumed that $\mathcal F(X)$, +the Ising critical point, it is nearly always assumed that $\fF(X)$, the universal scaling function, is an analytic function of $X$. However, it -has long been known that there exists an essential singularity in $\mathcal F$ +has long been known that there exists an essential singularity in $\fF$ at $X=0$, though its effects have long been believed to be unobservable \cite{fisher.1967.condensation}, or simply just neglected \cite{guida.1997.3dising,schofield.1969.parametric,schofield.1969.correlation,caselle.2001.critical,josephson.1969.equation,fisher.1999.trigonometric}. @@ -91,9 +119,9 @@ With careful analysis, we have found that assuming the presence of the essential singularity is predictive of the scaling form of e.g. the susceptibility. -The providence of the essential singularity can be understood using the +The provenance of the essential singularity can be understood using the methods of critical droplet theory for the decay of an Ising system in a -metastable state, i.e., an equilibrium Ising state for $T0$ +metastable state, i.e., an equilibrium Ising state for $T0$ subjected to a small negative external field $H<0$. The existence of an essential singularity has also been suggested by transfer matrix \cite{mccraw.1978.metastability,enting.1980.investigation} and \textsc{rg} @@ -112,63 +140,90 @@ the system that corresponds to decay. In critical droplet theory, the metastable state decays when a domain of the equilibrium state forms whose surface-energy cost for growth is outweighed by -bulk-energy gains. Assuming the free energy cost of the surface of the droplet -scales with the number of spins $N$ like $\Sigma N^\sigma$ and that of its -bulk scales like $-M|H|N$, the critical droplet size scales like -$N_\c\sim(M|H|/\Sigma)^{-1/(1-\sigma)}$ and the free energy of the critical -droplet scales like $\Delta -F_\c\sim\Sigma^{1/(1-\sigma)}(M|H|)^{-\sigma/(1-\sigma)}$. Assuming domains -have minimal surfaces, as evidenced by transfer matrix studies -\cite{gunther.1993.transfer-matrix}, $\sigma=1-\frac1d$ and $\Delta -F_\c\sim\Sigma^d(M|H|)^{-(d-1)}$. Assuming the singular scaling forms -$\Sigma=|g_t|^\mu\mathcal S(g_h|g_t|^{-\Delta})$ and $M=|g_t|^\beta\mathcal -M(g_h|g_t|^{-\Delta})$ and using known hyperscaling relations +bulk-energy gains. There is numerical evidence that, near the critical point, droplets are spherical \cite{gunther.1993.transfer-matrix}. The free energy cost of the surface of a droplet +scales with its radius $R$ like $\Sigma S_\dim R^{\dim-1}$ and that of its +bulk scales like $-M|H|V_\dim R^\dim$, where $S_\dim$ and $V_\dim$ are the surface area and +volume of a $(\dim-1)$-sphere, respectively, and $\Sigma$ is the surface tension of the equilibrium--metastable interface. The critical droplet size then is +$R_\c=(\dim-1)\Sigma/M|H|$ and the free energy of the critical +droplet is $\Delta +F_\c=\pi^{\dim/2}\Sigma^\dim((\dim-1)/M|H|)^{\dim-1}/\Gamma(1+\dim/2)$. +Assuming the typical singular scaling forms +$\Sigma/T=|t|^\mu\fS(h|t|^{-\Delta})$ and $M=|t|^\beta\mathcal +M(h|t|^{-\Delta})$ and using known hyperscaling relations \cite{widom.1981.interface}, this implies a scaling form \def\eqcritformone{ - \sim\mathcal S^d(g_h|g_t|^{-\Delta})(-g_h|g_t|^{-\Delta}\mathcal - M(g_h|g_t|^{-\Delta}))^{-(d-1)} + T\frac{\pi^{\dim/2}(\dim-1)^{\dim-1}}{\Gamma(1+\dim/2)}\frac{\fS^\dim(h|t|^{-\beta\delta})}{(-h|t|^{-\beta\delta}\fM(h|t|^{-\beta\delta}))^{\dim-1}} } \def\eqcritformtwo{ - \sim\mathcal G^{-(d-1)}(g_h|g_t|^{-\Delta}). + T\fG^{-(\dim-1)}(h|t|^{-\Delta}) } \ifreprint \[ \begin{aligned} - \Delta F_c&\eqcritformone - \\ - &\eqcritformtwo + \Delta F_\c + &=\eqcritformone\\ + &\sim\eqcritformtwo. \end{aligned} \] \else \[ - \Delta F_c\eqcritformone\eqcritformtwo + \Delta F_\c=\eqcritformone\sim\eqcritformtwo. \] \fi Since both surface tension and magnetization are finite and nonzero for $H=0$ -at $T1$ this function has an essential singularity in the invariant -combination $g_h|g_t|^{-\Delta}$. +combination $h|t|^{-\beta\delta}$. + +% $\Gamma/\Gamma_\sq=(D_\sq/D)(\fM(0)^{1/\beta}/\fM_\sq(0)^{1/\beta})^{-7/4}$ This form of $\im F$ for small $h$ is well known -\cite{langer.1967.condensation,harris.1984.metastability}. Henceforth we will -assume $h$ and $t$ are sufficiently small that $g_t\simeq t$, $g_h\simeq h$, -and all functions of both variables can be truncated at lowest order. We make -the scaling ansatz that the imaginary part of the metastable free energy has -the same singular behavior as the real part of the equilibrium free energy, -and that for small $t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\mathcal -H(h|t|^{-\Delta})$ for +\cite{langer.1967.condensation,harris.1984.metastability}. We make the scaling +ansatz that the imaginary part of the metastable free energy has the same +singular behavior as the real part of the equilibrium free energy, and that for +small $t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\fiF(h|t|^{-\beta\delta})$ for \[ - \mathcal H(X)=A\Theta(-X)(-X)^\zeta e^{-1/(-BX)^{d-1}}, + \fiF(X)=-A\Theta(-X)(-X)^be^{-1/(-BX)^{\dim-1}}, \label{eq:im.scaling} \] where $\Theta$ is the Heaviside function. Results from combining an analysis of fluctuations on the surface of critical droplets with \textsc{rg} recursion -relations suggest that $\zeta=-(d-3)d/2$ for $d=2,4$ and $\zeta=-7/3$ for +relations suggest that $b=-(d-3)d/2$ for $d=2,4$ and $b=-7/3$ for $d=3$ \cite{houghton.1980.metastable,rudnick.1976.equations,gunther.1980.goldstone}. Assuming that $F$ is analytic in the upper complex-$h$ plane, the real part of @@ -180,24 +235,24 @@ free energy using the Kramers--Kronig relation This relationship has been used to compute high-order moments of the free energy in $H$ in good agreement with transfer matrix expansions \cite{lowe.1980.instantons}. Here, we compute the integral to come to explicit -functional forms. In \textsc{3d} and \textsc{4d} this can be computed +functional forms. In \threedee and \fourdee this can be computed explicitly given our scaling ansatz, yielding \def\eqthreedeeone{ - \mathcal F^{\text{\textsc{3d}}}(X)&= - \frac{AB^{1/3}}{12\pi X^2}e^{-1/(BX)^2} - \bigg[\Gamma(\tfrac16)E_{7/6}(-(BX)^{-2}) + \fF^\threedee(Y/B)&= + \frac{A}{12}\frac{e^{-1/Y^2}}{Y^2} + \bigg[4Y\Gamma(\tfrac23)E_{5/3}(-Y^{-2}) } \def\eqthreedeetwo{ - -4BX\Gamma(\tfrac23)E_{5/3}(-(BX)^{-2})\bigg] + -\Gamma(\tfrac16)E_{7/6}(-Y^{-2})\bigg] } \def\eqfourdeeone{ - \mathcal F^{\text{\textsc{4d}}}(X)&= - -\frac{A}{9\pi X^2}e^{1/(BX)^3} - \Big[3\ei(-(BX)^{-3}) + \fF^\fourdee(Y/B)&= + \frac{A}{9\pi}\frac{e^{1/Y^3}}{Y^2} + \Big[3\ei(-Y^{-3}) } \def\eqfourdeetwo{ - +3\Gamma(\tfrac23)\Gamma(\tfrac13,(BX)^{-3}) - +\Gamma(\tfrac13)\Gamma(-\tfrac13,(BX)^{-3})\Big] + +3\Gamma(\tfrac23)\Gamma(\tfrac13,Y^{-3}) + +\Gamma(\tfrac13)\Gamma(-\tfrac13,Y^{-3})\Big] } \ifreprint \begin{align} @@ -211,62 +266,80 @@ explicitly given our scaling ansatz, yielding \eqfourdeeone \\ &\hspace{2em} - \eqfourdeetwo + \eqfourdeetwo. \end{aligned} \end{align} \else \begin{align} \eqthreedeeone\eqthreedeetwo \\ - \eqfourdeeone\eqfourdeetwo + \eqfourdeeone\eqfourdeetwo. \end{align} \fi -for \textsc{4d}. At the level of truncation we are working at, the Kramers--Kronig relation -does not converge in \textsc{2d}. However, the higher moments can still be +does not converge in \twodee. However, the higher moments can still be extracted, e.g., the susceptibility, by taking \[ - \chi\propto\pd[2]Fh + \chi=\pd[2]Fh =\frac2\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. \] -With $\chi=|t|^{-\gamma}\mathcal Y(h|t|^{-\Delta})$, this yields +With $\chi=|t|^{-\gamma}\fX(h|t|^{-\Delta})$, this yields \[ - \mathcal Y^{\text{\textsc{2d}}}(X)=\frac{C}{2(BX)^3}\big[BX(BX-1)-e^{1/BX}\ei(-1/BX)\big] + \fX^\twodee(Y/B)=\frac{AB^2}{\pi Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big] \label{eq:sus_scaling} \] -for some constant $C$. Previous work at zero field suggests that -$C=C_{0-}/T_c$, with $C_{0-}=0.025\,536\,971\,9$ -\cite{barouch.1973.susceptibility}. Scaling forms for the free energy can -then be extracted by integration and comparison with known exact results at -zero field, yielding -\[ - \mathcal M^{\text{\textsc{2d}}}(X)=\mathcal M(0)+D-\frac{D}{BX}(BX-1)e^{1/BX}\ei(-1/BX) +The constant $A$ can be fixed by zero-field results, with $\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=\frac{2AB^2}\pi$. +Scaling forms for the free energy can then be extracted by direct integration +and their constants of integration fixed by known zero field values, yielding +\begin{align} + \fM^\twodee(Y/B) + &=\fM(0)+\frac{ABT_c}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\ \label{eq:mag_scaling} -\] -with $\mathcal M(0)=\big(2(\sqrt2-1)\big)^{1/4}\big((4+3\sqrt2)\sinh^{-1}1\big)^{1/8}$ -\cite{onsager.1944.crystal} and $D$ constant, and -\[ - \mathcal F^{\text{\textsc{2d}}}(X)=\mathcal F(0)+EX(\mathcal M(0)+De^{1/BX}\ei(-1/BX)) -\] -for $\mathcal F(0)=?$ and $E$ constant. + \fF^\twodee(Y/B) + &=\fF(0)+T_cY\bigg(\frac{\fM(0)}B+\frac{AT_c}\pi e^{1/Y}\ei(-1/Y)\bigg) +\end{align} +with $F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})+t^{2-\alpha}\log|t|$ in two dimensions. + +Previous work at zero field suggests that +$\fX(0)=\frac{2AB^2}\pi=C_{0-}/T_\c$, with $C_{0-}=0.025\,536\,971\,9$ +\cite{barouch.1973.susceptibility}. +Exact results for the \twodee Ising +model have $\fS(0)=4$ and $\fM(0)=(2^{5/2}\arcsinh1)^\beta$, so +$B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\sinh^{-1}1)^{15/8})^{-1}$. How predictive are these scaling forms in the proximity of the critical point -and the abrupt transition line? We used a form of the Wolff algorithm modified -to remain efficient in the presence of an external field by incorporating the -field as another spin with coupling $|H|$ to all others -\cite{dimitrovic.1991.finite}. Data was then taken for susceptibility and -magnetization for $|t|,h\leq0.1$. This data is plotted in Figs.~\ref{fig:sus} +and the abrupt transition line? We simulated the \twodee Ising model on square, triangluar, and hexagonal lattices using a form of the Wolff algorithm modified +to remain efficient in the presence of an external field. Briefly, the external field $H$ is applied by adding an extra spin $s_0$ with coupling $|H|$ to all others +\cite{dimitrovic.1991.finite}. A quickly converging estimate for the magnetization in the finite-size system was then made by taking $M=\sgn(H)s_0\sum s_i$, i.e., the magnetization relative to the external spin. For the \twodee Ising model on a square lattice, exact results at zero temperature have $\fS(0)=4/T_c$, $\fM(0)=(2^{5/2}\arcsinh1)^\beta$ \cite{onsager.1944.crystal}, and $\fX(0)=C_{0-}/T_\c$ with $C_{0-}=0.025\,536\,971\,9$ +\cite{barouch.1973.susceptibility}, so that $B=\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$ and $A=\frac\pi2\fX(0)/B^2=2^{11/8}\pi^3(\arcsinh1)^{19/4}C_{0-}$. Data from other lattices can be mapped onto that of the square lattice using lattice--lattice scaling rules \cite{betts.1971.lattice}. If $D$ is the critical amplitude for the magnetization along the critical isotherm, i.e., $h=DM^\delta$ for $t=0$, then $t$ and $h$ for lattice $X$ are related to those of lattice $Y$ by $\frac{h_X}{D_X}=\frac{h_Y}{D_Y}$ and $t_X\fM_X(0)^{1/\beta}=t_Y\fM_Y(0)^{1/\beta}$. While $D$ is not known exactly, the ratios $D/D_\sq$ for any exactly solvable \twodee lattice are. These are tabulated, along with $\fM(0)$, in Table~\ref{tab:consts}. +Data was then taken for susceptibility and +magnetization for $T_\c-T,H\leq0.1$. This data is plotted in Figs.~\ref{fig:sus} and \ref{fig:mag}, along with collapses of data onto a single universal curve in the insets of those figures. As can be seen, there is very good agreement between our proposed functional forms and what is measured. +\begin{table} + \centering + \begin{tabular}{c|llc} + Lattice & $T_\c$ & $\fM(0)^{1/\beta}$ & $D/D_\sq$ \\ + \hline % ------------------------------------------------------------------- + Square & $2/\log(1+\sqrt2)$ & $2^{5/2}\arcsinh1$ & 1 \\ + Triangular & $4/\log3$ & $4\log3$ & $3^{3/2}/4$ \\ + Hexagonal & $2/\log(2+\sqrt3)$ & $\frac8{\sqrt3}\arccosh2$ & $3^{3/2}/8$ + \end{tabular} + \caption{ + The critical temperatures and amplitudes for the magetization along both the coexistence line and the critical isotherm, for three different lattices. + } + \label{tab:consts} +\end{table} + \begin{figure} \input{figs/fig-sus} \caption{ Fit of scaling form \eqref{eq:sus_scaling} to numeric data. Data with sampling error taken from Monte Carlo simulations of an $L=2048$ - square-lattice Ising model with $t=0.01,0.02,\ldots,0.1$ and - $h=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Solid line shows fitted form, + square-lattice Ising model with $T_\c-T=0.01,0.02,\ldots,0.1$ and + $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Solid line shows fitted form, with $C=0.0111\pm0.0023$ and $B=0.173\pm0.072$. } \label{fig:sus} @@ -277,9 +350,9 @@ between our proposed functional forms and what is measured. \caption{ Fit of scaling form \eqref{eq:mag_scaling} to numeric data. Data with sampling error taken from Monte Carlo simulations of an $L=2048$ - square-lattice Ising model with $t=0.01,0.02,\ldots,0.1$ and - $h=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Solid line shows fitted form, - with $\mathcal M(0)=1.21039\pm0.00031$, + square-lattice Ising model with $T_\c-T=0.01,0.02,\ldots,0.1$ and + $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Solid line shows fitted form, + with $\fM(0)=1.21039\pm0.00031$, $D=0.09400\pm0.00035$, and $B=0.0861\pm0.0010$. } \label{fig:mag} @@ -287,7 +360,7 @@ between our proposed functional forms and what is measured. We have used results from the properties of the metastable Ising ferromagnet and the analytic nature of the free energy to derive the universal scaling -functions for the free energy, and in \textsc{2d} the magnetization and +functions for the free energy, and in \twodee the magnetization and susceptibility, in the limit of small $t$ and $h$. Because of an essential singularity in these functions at $h=0$---the abrupt transition line---their form cannot be modified by analytic redefinition of control or thermodynamic diff --git a/figs/fig-mag.gplot b/figs/fig-mag.gplot index df67fa9..c5c00db 100644 --- a/figs/fig-mag.gplot +++ b/figs/fig-mag.gplot @@ -10,33 +10,35 @@ t(T) = abs((T - Tc) / T) h(T, H) = H / T X(T, H) = h(T, H) * t(T)**(-Delta) -B = 0.0861 -A = 0.094 -M0 = 1.21039 +B = 1 / (2**(27./16) * pi * asinh(1)**(15./8)) +C0 = 0.0255369719 +#A = 2**(11./8) * pi**3 * asinh(1)**(19./4) * C0 +A = pi / 2 * C0 / (B**2 * Tc) +M0 = (2 * (sqrt(2) - 1))**(1./4) * ((4 + 3 * sqrt(2)) * asinh(1))**(1./8) -func = "figs/fig-mag_data-func.dat" -num = "data/data_square-2048.dat" +func = "figs/fig-mag_scaling-func.dat" +num = "data/data_square-4096.dat" set samples 10000 set key off set xrange [-20:1200] set yrange [1.18:1.75] -set ylabel offset 1,0 '$Mt^{-\beta}$' -set xlabel '$ht^{-\Delta}$' +set ylabel offset 1,0 '$M|t|^{-\beta}$' +set xlabel '$h|t|^{-\Delta}$' -plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, func using (10**$1 / B):(M0 + 10**($2) * A) with linespoints pt 0 lw 2 lc rgb "black" +plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, func using (10**$1 / B):(M0 + 10**($2) * Tc * A * B) with linespoints pt 0 lw 2 lc rgb "black" set size 0.6,0.6 set origin 0.38,0.2 set xrange [-3:3] set yrange [1.18:1.75] -set ylabel offset 4,0 '\footnotesize$Mt^{-\beta}$' -set xlabel offset 0,0.5 '\footnotesize$ht^{-\Delta}$' +set ylabel offset 4,0 '\footnotesize$M|t|^{-\beta}$' +set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\Delta}$' set xtics format '\footnotesize$10^{%g}$' -2,2,3 set ytics format '\footnotesize %g' -plot num using (log10(X($2, $3))):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc 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+C3 = -0.402641 +lambda = 4.39547 + +func = "figs/fig-sus_scaling-func.dat" +num = "data/data_square-4096.dat" set samples 10000 set key off set xrange [-20:1200] -set yrange [-0.2:13] -set ylabel offset 1,0 '$\chi t^{\gamma}\times 10^3$' -set xlabel '$ht^{-\Delta}$' +set yrange [-0.5:13] +set ylabel offset 1,0 '$\chi|t|^{\gamma}\times 10^3$' +set xlabel '$h|t|^{-\Delta}$' -plot num using (X($2, $3)):(10**3 * $10 * t($2)**gamma):(10**3 * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, func using (10**$1 / B):(10**(3+$2) * A) with linespoints pt 0 lw 2 lc rgb "black" +plot num using (X($2, $3)):(10**3 * $10 * t($2)**gamma):(10**3 * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, func using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", func using (10**$1 / B):(10**(3+$2) * A * B**2 + 10**3 * (C1 * (10**$1) + C2 * (10**$1)**2 + C3 * (10**$1)**3) / (1 + (lambda * (10**$1))**4)) with lines lt 5 lw 1 lc rgb "red" set size 0.6,0.6 set origin 0.38,0.37 set xrange [-3:3] set yrange [-4.5:-1.5] -set ylabel offset 2.5,0 '\footnotesize$\chi t^\gamma$' -set xlabel offset 0,0.5 '\footnotesize$ht^{-\Delta}$' +set ylabel offset 2.5,0 '\footnotesize$\chi|t|^\gamma$' +set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\Delta}$' set xtics format '\footnotesize$10^{%g}$' -2,2,3 set ytics format '\footnotesize$10^{%g}$' -4,1,-2 -plot num using (log10(X($2, $3))):(log10($10 * t($2)**gamma)):($11 / $10):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, func using ($1-log10(B)):($2+log10(A)) with linespoints pt 0 lw 2 lc rgb "black" +plot num using (log10(X($2, $3))):(log10($10 * t($2)**gamma)):($11 / $10):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, func using ($1-log10(B)):($2+log10(A * B**2)) with linespoints pt 0 lw 2 lc rgb "black", func using ($1-log10(B)):(log10(10**$2 * A * B**2 + (C1 * (10**$1) + C2 * (10**$1)**2 + C3 * (10**$1)**3) / (1 + (lambda * (10**$1))**4))) with lines lt 5 lw 1 lc rgb "red" diff --git a/figs/fig-sus_scaling-func.dat b/figs/fig-sus_scaling-func.dat new file mode 100644 index 0000000..5e437f9 --- /dev/null +++ b/figs/fig-sus_scaling-func.dat @@ -0,0 +1,10001 @@ +-5. -0.18602574886192413 +-4.999 -0.20913119813695588 +-4.998 -0.20046604123735157 +-4.997 -0.19205105738757128 +-4.996 -0.18386790080579413 +-4.995 -0.20643608740337896 +-4.994 -0.19799415341720805 +-4.993 -0.18978514841699462 +-4.992 -0.1818052997439795 +-4.991 -0.20385479304682436 +-4.99 -0.19562123762010597 +-4.989 -0.18761811209055468 +-4.988 -0.2096261748169799 +-4.987 -0.20137400449635612 +-4.986 -0.19334993728808125 +-4.985 -0.1855389066702293 +-4.984 -0.20704378775524288 +-4.983 -0.19899947319539305 +-4.982 -0.19116868556010758 +-4.981 -0.1835516101909172 +-4.98 -0.20456161228466713 +-4.979 -0.1967173450946228 +-4.978 -0.1890831707056503 +-4.977 -0.21004412701222364 +-4.976 -0.20218385108511208 +-4.975 -0.19453333739614445 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-5.475154440807386 +4.979000000000001 -5.476154430302053 +4.98 -5.477154419820876 +4.981 -5.478154409363801 +4.982000000000001 -5.479154398930772 +4.9830000000000005 -5.480154388521734 +4.984 -5.4811543781366305 +4.984999999999999 -5.482154367775409 +4.986000000000001 -5.483154357438014 +4.987 -5.484154347124388 +4.9879999999999995 -5.485154336834479 +4.989000000000001 -5.486154326568234 +4.99 -5.487154316325594 +4.991 -5.488154306106506 +4.992000000000001 -5.48915429591092 +4.993 -5.490154285738777 +4.994 -5.491154275590025 +4.995000000000001 -5.492154265464612 +4.996 -5.493154255362481 +4.997 -5.49415424528358 +4.997999999999999 -5.495154235227856 +4.9990000000000006 -5.496154225195257 +5. -5.497154215185726 \ No newline at end of file -- cgit v1.2.3-70-g09d2 From 71d7a60444ef7242b570688ae3c3392821316899 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 7 Jul 2017 14:53:09 -0400 Subject: added newest data --- data/data_square-4096.dat | 3 --- 1 file changed, 3 deletions(-) diff --git a/data/data_square-4096.dat b/data/data_square-4096.dat index c20fc7f..92be755 100644 --- a/data/data_square-4096.dat +++ b/data/data_square-4096.dat @@ -231,7 +231,6 @@ 4096 2.219190000000000 0.003716270000000 -1.532958818722084 0.000024244385283 0.776375689436236 0.000027367504881 3.612753320617716 0.185791678044592 4.603481087013527 0.230118251980786 4096 2.219190000000000 0.005255600000000 -1.538351491405584 0.000024203011902 0.782289808173262 0.000025601376225 3.330290193448026 0.182445824050304 3.726231848340700 0.186311355792018 4096 2.219190000000000 0.004419420000000 -1.535446559064359 0.000023257555063 0.779192713316548 0.000024689591760 3.414601292368074 0.159108293397347 3.848041105783602 0.192324330524426 -<<<<<<< HEAD 4096 2.219190000000000 0.008838830000000 -1.549735822825833 0.000023786806372 0.793161835899090 0.000022041746961 3.118352541436563 0.155396508083611 2.677595455738619 0.133622932722448 4096 2.219190000000000 0.006250000000000 -1.541643294326231 0.000021360157718 0.785635919484961 0.000021797321264 3.249265670827936 0.163329002373242 3.383627627642942 0.169030759271938 4096 2.219190000000000 0.010511200000000 -1.554676931162415 0.000022544857291 0.797293150243588 0.000020227224855 2.774326612988058 0.145821780998698 2.233239040512960 0.111439021213091 @@ -499,5 +498,3 @@ 4096 2.169190000000000 0.100000000000000 -1.776586856986539 0.000017301959927 0.892306161649298 0.000007725835774 1.678614276597641 0.086615728714766 0.334696563508317 0.016701316813372 4096 2.169190000000000 0.084089600000000 -1.751026813424001 0.000018186478030 0.886546661997251 0.000008584388036 1.783003866430986 0.090877754421349 0.397259015174436 0.019860006244751 4096 2.169190000000000 0.059460400000000 -1.709594569428963 0.000016644003763 0.875744865054176 0.000008701084399 1.797628205611451 0.092147530400258 0.491282195923778 0.024555336261293 -======= ->>>>>>> 19b9e75413387000fb5628fafa1d031726c01fca -- cgit v1.2.3-70-g09d2 From 8dbdb3a4b883ae7c22a0e71163f3aa1a327fda09 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 7 Jul 2017 19:53:09 -0400 Subject: cleaned up the repo, put both figures in one --- data_square-4096.dat | 486 --------------------- essential-ising.tex | 86 ++-- figs/fig-mag.gplot | 44 -- figs/fig-mag_data-func.dat | 1001 -------------------------------------------- figs/fig-mag_data-num.dat | 500 ---------------------- figs/fig-sus.gplot | 46 -- figs/fig-sus_data-func.dat | 1001 -------------------------------------------- figs/fig-sus_data-num.dat | 500 ---------------------- figs/fig-susmag.gplot | 81 ++++ makefile | 2 +- 10 files changed, 129 insertions(+), 3618 deletions(-) delete mode 100644 data_square-4096.dat delete mode 100644 figs/fig-mag.gplot delete mode 100644 figs/fig-mag_data-func.dat delete mode 100644 figs/fig-mag_data-num.dat delete mode 100644 figs/fig-sus.gplot delete mode 100644 figs/fig-sus_data-func.dat delete mode 100644 figs/fig-sus_data-num.dat create mode 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a/essential-ising.tex +++ b/essential-ising.tex @@ -3,7 +3,7 @@ % Created by Jaron Kent-Dobias on Thu Apr 20 12:50:56 EDT 2017. % Copyright (c) 2017 Jaron Kent-Dobias. All rights reserved. % -\documentclass[aps,prl,preprint]{revtex4-1} +\documentclass[aps,prl,reprint]{revtex4-1} \usepackage[utf8]{inputenc} \usepackage{amsmath,amssymb,latexsym,mathtools,xifthen} @@ -292,9 +292,9 @@ The constant $A$ can be fixed by zero-field results, with $\chi(t,0)|t|^\gamma=\ Scaling forms for the free energy can then be extracted by direct integration and their constants of integration fixed by known zero field values, yielding \begin{align} + \label{eq:mag_scaling} \fM^\twodee(Y/B) &=\fM(0)+\frac{ABT_c}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\ - \label{eq:mag_scaling} \fF^\twodee(Y/B) &=\fF(0)+T_cY\bigg(\frac{\fM(0)}B+\frac{AT_c}\pi e^{1/Y}\ei(-1/Y)\bigg) \end{align} @@ -308,54 +308,62 @@ model have $\fS(0)=4$ and $\fM(0)=(2^{5/2}\arcsinh1)^\beta$, so $B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\sinh^{-1}1)^{15/8})^{-1}$. How predictive are these scaling forms in the proximity of the critical point -and the abrupt transition line? We simulated the \twodee Ising model on square, triangluar, and hexagonal lattices using a form of the Wolff algorithm modified +and the abrupt transition line? We simulated the \twodee Ising model on square lattice using a form of the Wolff algorithm modified to remain efficient in the presence of an external field. Briefly, the external field $H$ is applied by adding an extra spin $s_0$ with coupling $|H|$ to all others \cite{dimitrovic.1991.finite}. A quickly converging estimate for the magnetization in the finite-size system was then made by taking $M=\sgn(H)s_0\sum s_i$, i.e., the magnetization relative to the external spin. For the \twodee Ising model on a square lattice, exact results at zero temperature have $\fS(0)=4/T_c$, $\fM(0)=(2^{5/2}\arcsinh1)^\beta$ \cite{onsager.1944.crystal}, and $\fX(0)=C_{0-}/T_\c$ with $C_{0-}=0.025\,536\,971\,9$ -\cite{barouch.1973.susceptibility}, so that $B=\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$ and $A=\frac\pi2\fX(0)/B^2=2^{11/8}\pi^3(\arcsinh1)^{19/4}C_{0-}$. Data from other lattices can be mapped onto that of the square lattice using lattice--lattice scaling rules \cite{betts.1971.lattice}. If $D$ is the critical amplitude for the magnetization along the critical isotherm, i.e., $h=DM^\delta$ for $t=0$, then $t$ and $h$ for lattice $X$ are related to those of lattice $Y$ by $\frac{h_X}{D_X}=\frac{h_Y}{D_Y}$ and $t_X\fM_X(0)^{1/\beta}=t_Y\fM_Y(0)^{1/\beta}$. While $D$ is not known exactly, the ratios $D/D_\sq$ for any exactly solvable \twodee lattice are. These are tabulated, along with $\fM(0)$, in Table~\ref{tab:consts}. +\cite{barouch.1973.susceptibility}, so that $B=\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$ and $A=\frac\pi2\fX(0)/B^2=2^{11/8}\pi^3(\arcsinh1)^{19/4}C_{0-}$. Data was then taken for susceptibility and -magnetization for $T_\c-T,H\leq0.1$. This data is plotted in Figs.~\ref{fig:sus} -and \ref{fig:mag}, along with collapses of data onto a single universal curve +magnetization for $T_\c-T,H\leq0.1$. This data is plotted in +Fig.~\ref{fig:scaling_fits}, along with collapses of data onto a single universal curve in the insets of those figures. As can be seen, there is very good agreement between our proposed functional forms and what is measured. +However, there are systematic differences that can be seen most clearly in the +magnetization. Since our method is known to only be accurate for high moments +of the free energy, we should expect that low moments require corrections. +Therefore, we also fit those corrections of the form +\begin{align} + \fX^{\twodee\prime}(X)&=\fX^\twodee(X)+\sum_{n=1}^Nf_n(BX)\\ + \fM^{\twodee\prime}(X)&=\fM^\twodee(X)+\frac{T_\c}B\sum_{n=1}^NF_n(BX) +\end{align} +where $F_n'(x)=f_n(x)$ and +\begin{align} + f_n(x)&=\frac{C_nx^n}{1+(\lambda x)^{n+1}}\\ + F_n(x)&=\frac{C_n\lambda^{-(n+1)}}{n+1}\log(1+(\lambda x)^{n+1}) +\end{align} +We fit these functions to our numeric data for $N=3$. The resulting curves are +also plotted in Fig.~\ref{fig:scaling_fits} as a dashed line. -\begin{table} - \centering - \begin{tabular}{c|llc} - Lattice & $T_\c$ & $\fM(0)^{1/\beta}$ & $D/D_\sq$ \\ - \hline % ------------------------------------------------------------------- - Square & $2/\log(1+\sqrt2)$ & $2^{5/2}\arcsinh1$ & 1 \\ - Triangular & $4/\log3$ & $4\log3$ & $3^{3/2}/4$ \\ - Hexagonal & $2/\log(2+\sqrt3)$ & $\frac8{\sqrt3}\arccosh2$ & $3^{3/2}/8$ - \end{tabular} - \caption{ - The critical temperatures and amplitudes for the magetization along both the coexistence line and the critical isotherm, for three different lattices. - } - \label{tab:consts} -\end{table} -\begin{figure} - \input{figs/fig-sus} - \caption{ - Fit of scaling form \eqref{eq:sus_scaling} to numeric data. Data with - sampling error taken from Monte Carlo simulations of an $L=2048$ - square-lattice Ising model with $T_\c-T=0.01,0.02,\ldots,0.1$ and - $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Solid line shows fitted form, - with $C=0.0111\pm0.0023$ and $B=0.173\pm0.072$. - } - \label{fig:sus} -\end{figure} + + +%\begin{table} +% \centering +% \begin{tabular}{c|llc} +% Lattice & $T_\c$ & $\fM(0)^{1/\beta}$ & $D/D_\sq$ \\ +% \hline % ------------------------------------------------------------------- +% Square & $2/\log(1+\sqrt2)$ & $2^{5/2}\arcsinh1$ & 1 \\ +% Triangular & $4/\log3$ & $4\log3$ & $3^{3/2}/4$ \\ +% Hexagonal & $2/\log(2+\sqrt3)$ & $\frac8{\sqrt3}\arccosh2$ & $3^{3/2}/8$ +% \end{tabular} +% \caption{ +% The critical temperatures and amplitudes for the magetization along both the coexistence line and the critical isotherm, for three different lattices. +% } +% \label{tab:consts} +%\end{table} \begin{figure} - \input{figs/fig-mag} + \input{figs/fig-susmag} \caption{ - Fit of scaling form \eqref{eq:mag_scaling} to numeric data. Data with - sampling error taken from Monte Carlo simulations of an $L=2048$ - square-lattice Ising model with $T_\c-T=0.01,0.02,\ldots,0.1$ and - $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Solid line shows fitted form, - with $\fM(0)=1.21039\pm0.00031$, - $D=0.09400\pm0.00035$, and $B=0.0861\pm0.0010$. + Comparisons of scaling forms \eqref{eq:sus_scaling} and + \eqref{eq:mag_scaling} to numeric data. Data with + sampling error taken from Monte Carlo simulations of a $4096\times4096$ + square-lattice Ising model with periodic boundary conditions and $T_\c-T=0.01,0.02,\ldots,0.1$ and + $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. The solid lines show our + analytic results, while the dashed lines have polynomial corrections of + the form \eqref{eq:poly} fit to the data for $N=3$, with $C_1=-0.00368$, + $C_2=-0.0191$, $C_3=0.0350$, and $\lambda=2.42$. } - \label{fig:mag} + \label{fig:scaling_fits} \end{figure} We have used results from the properties of the metastable Ising ferromagnet diff --git a/figs/fig-mag.gplot b/figs/fig-mag.gplot deleted file mode 100644 index c5c00db..0000000 --- a/figs/fig-mag.gplot +++ /dev/null @@ -1,44 +0,0 @@ - -set terminal pslatex rotate size 3.417,2.111 -set multiplot - -Tc = 2 / log(1 + sqrt(2)) -Delta = 15. / 8 -beta = 1. / 8 - -t(T) = abs((T - Tc) / T) -h(T, H) = H / T -X(T, H) = h(T, H) * t(T)**(-Delta) - -B = 1 / (2**(27./16) * pi * asinh(1)**(15./8)) -C0 = 0.0255369719 -#A = 2**(11./8) * pi**3 * asinh(1)**(19./4) * C0 -A = pi / 2 * C0 / (B**2 * Tc) -M0 = (2 * (sqrt(2) - 1))**(1./4) * ((4 + 3 * sqrt(2)) * asinh(1))**(1./8) - -func = "figs/fig-mag_scaling-func.dat" -num = "data/data_square-4096.dat" - -set samples 10000 -set key off - -set xrange [-20:1200] -set yrange [1.18:1.75] -set ylabel offset 1,0 '$M|t|^{-\beta}$' -set xlabel '$h|t|^{-\Delta}$' - -plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, func using (10**$1 / B):(M0 + 10**($2) * Tc * A * B) with linespoints pt 0 lw 2 lc rgb "black" - -set size 0.6,0.6 -set origin 0.38,0.2 -set xrange [-3:3] -set yrange [1.18:1.75] -set ylabel offset 4,0 '\footnotesize$M|t|^{-\beta}$' -set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\Delta}$' -set xtics format '\footnotesize$10^{%g}$' -2,2,3 -set ytics format '\footnotesize %g' - -plot num using (log10(X($2, $3))):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, func using ($1-log10(B)):(M0 + Tc * A * B * 10**$2) with linespoints pt 0 lw 2 lc black - - - diff --git a/figs/fig-mag_data-func.dat b/figs/fig-mag_data-func.dat deleted file mode 100644 index 29653f0..0000000 --- a/figs/fig-mag_data-func.dat +++ /dev/null @@ -1,1001 +0,0 @@ --5. -4.698976620880227 --4.99 -4.688976675606296 --4.98 -4.678976738233927 --4.97 -4.668977103542201 --4.96 -4.658977175055251 --4.95 -4.64897725444246 --4.94 -4.6389776168638575 --4.93 -4.628977705544413 --4.92 -4.618977802145736 --4.91 -4.608977906594962 --4.9 -4.598978269856612 --4.89 -4.588978384162751 --4.88 -4.578978506414977 --4.87 -4.568978870802035 --4.86 -4.558979003497131 --4.85 -4.548979144167584 --4.84 -4.538979511480103 --4.83 -4.528979663220596 --4.82 -4.518979823098883 --4.81 -4.508980093110478 --4.8 -4.498980366682533 --4.79 -4.488980546555846 --4.78 -4.478980829968869 --4.77 -4.468981024367286 --4.76 -4.45898131817752 --4.75 -4.448981616365706 --4.74 -4.438981832304746 --4.73 -4.428982141813062 --4.72 -4.41898245622204 --4.71 -4.408982694750725 --4.7 -4.398983021480837 --4.69 -4.38898327631837 --4.68 -4.378983615975297 --4.67 -4.368983961528195 --4.66 -4.3589842409399875 --4.65 -4.3489846005174 --4.64 -4.338984897588365 --4.63 -4.328985271917661 --4.62 -4.318985653142753 --4.61 -4.30898597712615 --4.6 -4.298986374275912 --4.59 -4.288986717536211 --4.58 -4.278987131460024 --4.57 -4.268987553391627 --4.56 -4.258987926201059 --4.55 -4.248988366205847 --4.54 -4.238988814948243 --4.53 -4.228989219150657 --4.52 -4.218989687352525 --4.51 -4.208990139510039 --4.5 -4.198990602608149 --4.49 -4.188991076885322 --4.48 -4.178991562587806 --4.47 -4.168992083269395 --4.46 -4.158992592054487 --4.45 -4.14899311305065 --4.4399999999999995 -4.138993646517517 --4.43 -4.128994214017157 --4.42 -4.11899477281752 --4.41 -4.10899534496253 --4.4 -4.098995930752351 --4.39 -4.088996549873465 --4.38 -4.078997163435489 --4.37 -4.068997791591377 --4.36 -4.058998434672682 --4.35 -4.048999092999943 --4.34 -4.038999784226322 --4.33 -4.029000473731878 --4.32 -4.019001179564121 --4.31 -4.009001902084939 --4.3 -3.9990026574533086 --4.29 -3.9890034141655693 --4.28 -3.9790041887463223 --4.27 -3.969004981596725 --4.26 -3.9590057931477527 --4.25 -3.949006637866626 --4.24 -3.939007487773016 --4.23 -3.929008357692256 --4.22 -3.9190092480716765 --4.21 -3.909010165801212 --4.2 -3.899011104656105 --4.1899999999999995 -3.889012059012884 --4.18 -3.879013041771961 --4.17 -3.8690140413498035 --4.16 -3.8590150643886725 --4.15 -3.8490161170082184 --4.14 -3.8390171884668987 --4.13 -3.8290182903827445 --4.12 -3.8190194125471444 --4.11 -3.8090205660956147 --4.1 -3.799021741331879 --4.09 -3.789022948958459 --4.08 -3.779024179771146 --4.07 -3.769025444034952 --4.06 -3.7590267330222162 --4.05 -3.7490280521725925 --4.04 -3.7390294065078074 --4.03 -3.7290307879815963 --4.02 -3.719032205884891 --4.01 -3.7090336526107213 --4. -3.699035137097331 --3.99 -3.689036652139535 --3.98 -3.679038206348264 --3.9699999999999998 -3.669039792916126 --3.96 -3.659041420145048 --3.95 -3.6490430816057717 --3.94 -3.639044781858449 --3.9299999999999997 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--3.26 -2.9593276218370663 --3.25 -2.9493359428995682 --3.24 -2.9393444572755207 --3.23 -2.929353169623009 --3.2199999999999998 -2.9193620844433785 --3.21 -2.909371206351528 --3.2 -2.8993805403120763 --3.19 -2.8893900910799566 --3.1799999999999997 -2.879399863837082 --3.17 -2.869409863648785 --3.16 -2.859420095704796 --3.15 -2.849430565532074 --3.1399999999999997 -2.8394412785648133 --3.13 -2.829452240370517 --3.12 -2.819463456847051 --3.11 -2.809474933758619 --3.0999999999999996 -2.7994866772662554 --3.09 -2.7894986934758235 --3.08 -2.779510988640467 --3.0700000000000003 -2.7695235693382174 --3.06 -2.759536442114671 --3.05 -2.7495496136729214 --3.04 -2.73956309103892 --3.0300000000000002 -2.7295768811752987 --3.02 -2.7195909914236975 --3.01 -2.7096054291312823 --3. -2.699620201844405 --2.9899999999999998 -2.689635317333078 --2.98 -2.6796507835196555 --2.9699999999999998 -2.6696666085307026 --2.96 -2.659682800609395 --2.95 -2.6496993682585663 --2.94 -2.639716320151525 --2.93 -2.6297336651607974 --2.92 -2.6197514123815897 --2.91 -2.609769571055506 --2.9 -2.59978815069819 --2.89 -2.5898071610409876 --2.88 -2.5798266119785587 --2.87 -2.5698465136915165 --2.86 -2.5598668765719665 --2.85 -2.5498877112671097 --2.84 -2.539909028613332 --2.83 -2.5299308397484213 --2.82 -2.519953156044999 --2.81 -2.5099759891510587 --2.8 -2.4999993509305676 --2.79 -2.4900232535671973 --2.78 -2.480047709504156 --2.77 -2.470072731482451 --2.76 -2.460098332487579 --2.75 -2.450124525845017 --2.7399999999999998 -2.440151325166729 --2.73 -2.430178744386789 --2.7199999999999998 -2.4202067977142354 --2.71 -2.4102354997207076 --2.6999999999999997 -2.4002648652996363 --2.69 -2.3902949096623476 --2.68 -2.3803256483886286 --2.67 -2.370357097385258 --2.66 -2.360389272942012 --2.65 -2.35042219169209 --2.64 -2.3404558706661645 --2.63 -2.3304903272556396 --2.62 -2.3205255792649755 --2.61 -2.310561644876933 --2.6 -2.3005985427034865 --2.59 -2.290636291753244 --2.58 -2.2806749114810807 --2.57 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-gamma = 7. / 4 - -t(T) = abs((T - Tc) / T) -h(T, H) = H / T -X(T, H) = h(T, H) * t(T)**(-Delta) - -B = 1 / (2**(27./16) * pi * asinh(1)**(15./8)) -A = 0.0255369719 / Tc * pi / (2 * B**2) - -C1 = -0.0160968 -C2 = 0.116242 -C3 = -0.402641 -lambda = 4.39547 - -func = "figs/fig-sus_scaling-func.dat" -num = "data/data_square-4096.dat" - -set samples 10000 -set key off - -set xrange [-20:1200] -set yrange [-0.5:13] -set ylabel offset 1,0 '$\chi|t|^{\gamma}\times 10^3$' -set xlabel '$h|t|^{-\Delta}$' - -plot num using (X($2, $3)):(10**3 * $10 * t($2)**gamma):(10**3 * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, func using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", func using (10**$1 / B):(10**(3+$2) * A * B**2 + 10**3 * (C1 * (10**$1) + C2 * (10**$1)**2 + C3 * (10**$1)**3) / (1 + (lambda * (10**$1))**4)) with lines lt 5 lw 1 lc rgb "red" - -set size 0.6,0.6 -set origin 0.38,0.37 -set xrange [-3:3] -set yrange [-4.5:-1.5] -set ylabel offset 2.5,0 '\footnotesize$\chi|t|^\gamma$' -set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\Delta}$' -set xtics format '\footnotesize$10^{%g}$' -2,2,3 -set ytics format '\footnotesize$10^{%g}$' -4,1,-2 - -plot num using (log10(X($2, $3))):(log10($10 * t($2)**gamma)):($11 / $10):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, func using ($1-log10(B)):($2+log10(A * B**2)) with linespoints pt 0 lw 2 lc rgb "black", func using ($1-log10(B)):(log10(10**$2 * A * B**2 + (C1 * (10**$1) + C2 * (10**$1)**2 + C3 * (10**$1)**3) / (1 + (lambda * (10**$1))**4))) with lines lt 5 lw 1 lc rgb "red" - - - diff --git a/figs/fig-sus_data-func.dat b/figs/fig-sus_data-func.dat deleted file mode 100644 index 972c77b..0000000 --- a/figs/fig-sus_data-func.dat +++ /dev/null @@ -1,1001 +0,0 @@ --5. 0.010094128168228536 --4.99 0.0004986394100467353 --4.98 -0.008441735254514486 --4.97 0.010954806238037133 --4.96 0.002527852467923256 --4.95 -0.005308016370920201 --4.94 0.0114293828525401 --4.93 0.004018232985220573 --4.92 -0.002858226194217986 --4.91 -0.009219894501557 --4.9 0.005081385417224557 --4.89 -0.0009618799060500723 --4.88 -0.006542816311444805 --4.87 0.005798926687913552 --4.86 0.0004888958038767844 --4.85 -0.004414469974308102 --4.84 0.006246050655940169 --4.83 0.0015752929121863565 --4.82 -0.0027318837493295303 --4.81 -0.00005661662281440823 --4.8 0.002370862728926167 --4.79 -0.0014169159468057889 --4.78 0.0008653476550368721 --4.77 -0.0025424410197080745 --4.76 -0.0003992250773220128 --4.75 0.0015469120819587867 --4.74 -0.001449908548519092 --4.73 0.00037816511803153026 --4.72 0.0020366851344616396 --4.71 -0.000600885400499167 --4.7 0.0009588492800455502 --4.69 -0.0014114510648439034 --4.68 0.000051240900424078435 --4.67 0.001381830333725127 --4.66 -0.0007050759037017436 --4.65 0.0005442342992692582 --4.64 -0.0013309799160835763 --4.63 -0.00015920310383799646 --4.62 0.00090685946257924 --4.61 -0.0007436599033119168 --4.6 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/dev/null +++ b/figs/fig-susmag.gplot @@ -0,0 +1,81 @@ + +set terminal pslatex rotate size 3.417,4.222 +set multiplot + +Tc = 2 / log(1 + sqrt(2)) +Delta = 15. / 8 +gamma = 7. / 4 +beta = 1. / 8 + +t(T) = abs((T - Tc) / T) +h(T, H) = H / T +X(T, H) = h(T, H) * t(T)**(-Delta) + +poly(A, l, m, X) = A * X**m / (1 + (l * X)**(m + 1)) +polyint(A, l, m, X) = A * l**(-(m + 1)) * log(1 + (l * X)**(m + 1)) / (m + 1) + +M0 = (2**2.5 * asinh(1))**0.125 +B = Tc**2 * M0 / (16 * pi) +C0 = 0.0255369719 +A = pi / 2 * C0 / (B**2 * Tc) + +C1 = -0.00368218 +C2 = -0.0190753 +C3 = 0.0349601 +lamb = 2.41718 + +susfunc = "figs/fig-sus_scaling-func.dat" +magfunc = "figs/fig-mag_scaling-func.dat" + +num = "data/data_square-4096.dat" + +set samples 10000 +set key off + +set size 1,1 - 0.52 +set origin 0,0.52 +set xrange [-20:1200] +set yrange [-0.5:13] +set ylabel offset 1,0 '$\chi|t|^{\gamma}\times 10^3$' +set lmargin 6 +set xtics format '' +set bmargin 0.2 + +plot num using (X($2, $3)):(10**3 * $10 * t($2)**gamma):(10**3 * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B):(10**(3+$2) * A * B**2 + 10**3 * (poly(C1, lamb, 1, 10**$1) + poly(C2, lamb, 2, 10**$1) + poly(C3, lamb, 3, 10**$1))) with lines dt 2 lw 2 lc black + +set bmargin -1 +set tmargin 0.2 +set size 1,0.52 +set origin 0,0 +set xrange [-20:1200] +set yrange [1.18:1.75] +set ylabel offset 1,0 '$M|t|^{-\beta}$' +set xlabel '$h|t|^{-\Delta}$' +set xtics format '%g' + +plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A * B) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A * B + (polyint(Tc * C1, lamb, 1, 10**$1) + polyint(Tc * C2, lamb, 2, 10**$1) + polyint(Tc * C3, lamb, 3, 10**$1)) / B) smooth csplines with lines dt 2 lw 2 lc black + +set tmargin -1 +set lmargin -1 +set size 0.65,0.325 +set origin 0.31,0.5 + 0.29 / 2 +set xrange [-3:3] +set yrange [-4.5:-1.5] +set ylabel offset 2.5,0 '\footnotesize$\chi|t|^\gamma$' +set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\Delta}$' +set xtics format '\footnotesize$10^{%g}$' -2,2,3 +set ytics format '\footnotesize$10^{%g}$' -4,1,-2 + +plot num using (log10(X($2, $3))):(log10($10 * t($2)**gamma)):($11 / $10):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using ($1-log10(B)):($2+log10(A * B**2)) with linespoints pt 0 lw 2 lc rgb "black", susfunc using ($1-log10(B)):(log10(10**$2 * A * B**2 + poly(C1, lamb, 1, 10**$1) + poly(C2, lamb, 2, 10**$1) + poly(C3, lamb, 3, 10**$1))) with lines dt 2 lw 2 lc black + +set size 0.65,0.325 +set origin 0.31,0.23 / 2 +set xrange [-3:3] +set yrange [1.18:1.75] +set ylabel offset 4,0 '\footnotesize$M|t|^{-\beta}$' +set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\Delta}$' +set xtics format '\footnotesize$10^{%g}$' -2,2,3 +set ytics format '\footnotesize {%g}' 1.2,.2,1.6 + +plot num using (log10(X($2, $3))):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using ($1-log10(B)):(M0 + Tc * A * B * 10**$2) with linespoints pt 0 lw 2 lc black, magfunc using ($1 - log10(B)):(M0 + 10**($2) * Tc * A * B + (polyint(Tc * C1, lamb, 1, 10**$1) + polyint(Tc * C2, lamb, 2, 10**$1) + polyint(Tc * C3, lamb, 3, 10**$1)) / B) with lines dt 2 lw 2 lc black + diff --git a/makefile b/makefile index 7f43e44..8913449 100644 --- a/makefile +++ b/makefile @@ -1,6 +1,6 @@ DOC=essential-ising -FIGS=fig-sus fig-mag +FIGS=fig-susmag all: ${FIGS:%=figs/%.tex} rubber $(DOC).tex -- cgit v1.2.3-70-g09d2 From 789e65eaf0be44731d36168e8fc2e966ca7a4516 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 7 Jul 2017 20:07:35 -0400 Subject: added figure dependencies to the makefile --- makefile | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/makefile b/makefile index 8913449..ffa4c63 100644 --- a/makefile +++ b/makefile @@ -1,12 +1,14 @@ DOC=essential-ising FIGS=fig-susmag +DATA=data_square-4096 +FUNCS=fig-mag_scaling-func fig-sus_scaling-func all: ${FIGS:%=figs/%.tex} rubber $(DOC).tex dvipdf $(DOC).dvi -figs/%.tex: figs/%.gplot +figs/%.tex: figs/%.gplot ${DATA:%=data/%.dat} ${FUNCS:%=figs/%.dat} gnuplot $< > $@ clean: -- cgit v1.2.3-70-g09d2 From 5e1b43e450537f16f220b71409a0f133a1008302 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 7 Jul 2017 20:44:46 -0400 Subject: reference fix --- essential-ising.tex | 13 ++++++++----- 1 file changed, 8 insertions(+), 5 deletions(-) diff --git a/essential-ising.tex b/essential-ising.tex index 90c42b9..819775b 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -311,7 +311,7 @@ How predictive are these scaling forms in the proximity of the critical point and the abrupt transition line? We simulated the \twodee Ising model on square lattice using a form of the Wolff algorithm modified to remain efficient in the presence of an external field. Briefly, the external field $H$ is applied by adding an extra spin $s_0$ with coupling $|H|$ to all others \cite{dimitrovic.1991.finite}. A quickly converging estimate for the magnetization in the finite-size system was then made by taking $M=\sgn(H)s_0\sum s_i$, i.e., the magnetization relative to the external spin. For the \twodee Ising model on a square lattice, exact results at zero temperature have $\fS(0)=4/T_c$, $\fM(0)=(2^{5/2}\arcsinh1)^\beta$ \cite{onsager.1944.crystal}, and $\fX(0)=C_{0-}/T_\c$ with $C_{0-}=0.025\,536\,971\,9$ -\cite{barouch.1973.susceptibility}, so that $B=\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$ and $A=\frac\pi2\fX(0)/B^2=2^{11/8}\pi^3(\arcsinh1)^{19/4}C_{0-}$. +\cite{barouch.1973.susceptibility}, so that $B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$ and $A=\frac\pi2\fX(0)/B^2=2^{11/8}\pi^3(\arcsinh1)^{19/4}C_{0-}$. Data was then taken for susceptibility and magnetization for $T_\c-T,H\leq0.1$. This data is plotted in Fig.~\ref{fig:scaling_fits}, along with collapses of data onto a single universal curve @@ -326,10 +326,13 @@ Therefore, we also fit those corrections of the form \fM^{\twodee\prime}(X)&=\fM^\twodee(X)+\frac{T_\c}B\sum_{n=1}^NF_n(BX) \end{align} where $F_n'(x)=f_n(x)$ and -\begin{align} - f_n(x)&=\frac{C_nx^n}{1+(\lambda x)^{n+1}}\\ - F_n(x)&=\frac{C_n\lambda^{-(n+1)}}{n+1}\log(1+(\lambda x)^{n+1}) -\end{align} +\[ + \begin{aligned} + f_n(x)&=\frac{C_nx^n}{1+(\lambda x)^{n+1}}\\ + F_n(x)&=\frac{C_n\lambda^{-(n+1)}}{n+1}\log(1+(\lambda x)^{n+1}) + \end{aligned} + \label{eq:poly} +\] We fit these functions to our numeric data for $N=3$. The resulting curves are also plotted in Fig.~\ref{fig:scaling_fits} as a dashed line. -- cgit v1.2.3-70-g09d2 From a08ce3d79426725955923c486ffe8127493682ca Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Sat, 8 Jul 2017 00:58:43 -0400 Subject: many small edits --- essential-ising.bib | 11 +++++++ essential-ising.tex | 90 +++++++++++++++++++++++++++------------------------ figs/fig-susmag.gplot | 6 ++-- makefile | 8 +++-- 4 files changed, 66 insertions(+), 49 deletions(-) diff --git a/essential-ising.bib b/essential-ising.bib index 3e39487..556c4e7 100644 --- a/essential-ising.bib +++ b/essential-ising.bib @@ -406,3 +406,14 @@ publisher={APS} } +@article{zinn.1996.universal, + title={Universal surface-tension and critical-isotherm amplitude ratios in three dimensions}, + author={Zinn, Shun-yong and Fisher, Michael E}, + journal={Physica A: Statistical Mechanics and its Applications}, + volume={226}, + number={1-2}, + pages={168--180}, + year={1996}, + publisher={Elsevier} +} + diff --git a/essential-ising.tex b/essential-ising.tex index 819775b..a64f75b 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -64,10 +64,10 @@ \begin{document} -\title{Essential Singularities in the Ising Universal Scaling Functions} +\title{Essential Singularities in Universal Scaling Functions at the Ising Coexistence Line} \author{Jaron Kent-Dobias} \author{James P.~Sethna} -\affiliation{Cornell University} +\affiliation{Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA} \date\today @@ -76,7 +76,7 @@ used to construct a scaling ansatz for the imaginary component of the free energy of an Ising model in its metastable state close to the critical point. The analytic properties of the free energy are used to determine - asymptotic scaling functions for the free energy in the vicinity of the + scaling functions for the free energy in the vicinity of the critical point and the abrupt transition line. These functions have essential singularities at zero field. Analogous forms for the magnetization and susceptibility in two-dimensions are fit to numeric data and show good @@ -102,13 +102,7 @@ thermodynamic variables. \textsc{Rg} analysis predicts that the singular part of the free energy per site $F$ as a function of reduced temperature $t=1-T_\c/T$ and field $h=H/T$ in the vicinity of the critical point takes the scaling form -$F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\Delta})$ \footnote{Technically -we should write $\fF_{\pm}$ to indicate that the universal scaling -function takes a different form for $t<0$ and $t>0$, but we will restrict -ourselves entirely to $t<0$ and hence $\fF_-$ for the purposes of this -paper.}, where $\Delta=\beta\delta$ and $t$, $h$ are analytic functions of -$t$, $h$ that transform exactly linearly under \textsc{rg} -\cite{cardy.1996.scaling,aharony.1983.fields}. When studying the properties of +$F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})$ for the low temperature phase $t<0$ \cite{cardy.1996.scaling,aharony.1983.fields}. When studying the properties of the Ising critical point, it is nearly always assumed that $\fF(X)$, the universal scaling function, is an analytic function of $X$. However, it has long been known that there exists an essential singularity in $\fF$ @@ -140,22 +134,24 @@ the system that corresponds to decay. In critical droplet theory, the metastable state decays when a domain of the equilibrium state forms whose surface-energy cost for growth is outweighed by -bulk-energy gains. There is numerical evidence that, near the critical point, droplets are spherical \cite{gunther.1993.transfer-matrix}. The free energy cost of the surface of a droplet -scales with its radius $R$ like $\Sigma S_\dim R^{\dim-1}$ and that of its -bulk scales like $-M|H|V_\dim R^\dim$, where $S_\dim$ and $V_\dim$ are the surface area and +bulk-energy gains. There is numerical evidence that, near the critical point, +droplets are spherical \cite{gunther.1993.transfer-matrix}. The free energy +cost of the surface of a droplet of radius $R$ +is $\Sigma S_\dim R^{\dim-1}$ and that of its +bulk is $-M|H|V_\dim R^\dim$, where $S_\dim$ and $V_\dim$ are the surface area and volume of a $(\dim-1)$-sphere, respectively, and $\Sigma$ is the surface tension of the equilibrium--metastable interface. The critical droplet size then is $R_\c=(\dim-1)\Sigma/M|H|$ and the free energy of the critical droplet is $\Delta F_\c=\pi^{\dim/2}\Sigma^\dim((\dim-1)/M|H|)^{\dim-1}/\Gamma(1+\dim/2)$. Assuming the typical singular scaling forms -$\Sigma/T=|t|^\mu\fS(h|t|^{-\Delta})$ and $M=|t|^\beta\mathcal -M(h|t|^{-\Delta})$ and using known hyperscaling relations +$\Sigma/T=|t|^\mu\fS(h|t|^{-\beta\delta})$ and $M=|t|^\beta\mathcal +M(h|t|^{-\beta\delta})$ and using known hyperscaling relations \cite{widom.1981.interface}, this implies a scaling form \def\eqcritformone{ T\frac{\pi^{\dim/2}(\dim-1)^{\dim-1}}{\Gamma(1+\dim/2)}\frac{\fS^\dim(h|t|^{-\beta\delta})}{(-h|t|^{-\beta\delta}\fM(h|t|^{-\beta\delta}))^{\dim-1}} } \def\eqcritformtwo{ - T\fG^{-(\dim-1)}(h|t|^{-\Delta}) + T\fG^{-(\dim-1)}(h|t|^{-\beta\delta}) } \ifreprint \[ @@ -181,7 +177,7 @@ and the critical domain radius $R_c$, with Bh|t|^{-\beta\delta}=\bigg(\frac{\Gamma(1+\dim/2)}{\pi^{\dim/2}\fS(0)(\xi_0^-)^{\dim-1}}\bigg)^{1/(\dim-1)}\frac\xi{R_\c} \] where $\xi=\xi_0^-|t|^{-\nu}$ for $t1$ this function has an essential singularity in the invariant combination $h|t|^{-\beta\delta}$. @@ -283,7 +279,7 @@ extracted, e.g., the susceptibility, by taking \chi=\pd[2]Fh =\frac2\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. \] -With $\chi=|t|^{-\gamma}\fX(h|t|^{-\Delta})$, this yields +With $\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, this yields \[ \fX^\twodee(Y/B)=\frac{AB^2}{\pi Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big] \label{eq:sus_scaling} @@ -300,12 +296,18 @@ and their constants of integration fixed by known zero field values, yielding \end{align} with $F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})+t^{2-\alpha}\log|t|$ in two dimensions. -Previous work at zero field suggests that -$\fX(0)=\frac{2AB^2}\pi=C_{0-}/T_\c$, with $C_{0-}=0.025\,536\,971\,9$ -\cite{barouch.1973.susceptibility}. -Exact results for the \twodee Ising -model have $\fS(0)=4$ and $\fM(0)=(2^{5/2}\arcsinh1)^\beta$, so -$B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\sinh^{-1}1)^{15/8})^{-1}$. +How are these functional forms to be interpreted? They are not asymptotic +forms in any sense, as there is no limit of $t$ or $h$ in which they become +aribitrarily correct. It is well established that this method of using +unphysical or metastable elements of a theory to extract properties of the +stable or equilibrium theory only works for high moments of those predictions +\cite{parisi.1977.asymptotic,bogomolny.1977.dispersion}. +These functions should be understood as possessing exactly the correct +singularity at the coexistence line, but requiring polynomial corrections, +especially for smaller integer powers. Using these forms in conjunction with +existing methods of describing the critical equation of state or critical +properties with analytic functions in $h$ will incorporate these low-order +corrections while preserving the correct singular structure. How predictive are these scaling forms in the proximity of the critical point and the abrupt transition line? We simulated the \twodee Ising model on square lattice using a form of the Wolff algorithm modified @@ -313,31 +315,30 @@ to remain efficient in the presence of an external field. Briefly, the external \cite{dimitrovic.1991.finite}. A quickly converging estimate for the magnetization in the finite-size system was then made by taking $M=\sgn(H)s_0\sum s_i$, i.e., the magnetization relative to the external spin. For the \twodee Ising model on a square lattice, exact results at zero temperature have $\fS(0)=4/T_c$, $\fM(0)=(2^{5/2}\arcsinh1)^\beta$ \cite{onsager.1944.crystal}, and $\fX(0)=C_{0-}/T_\c$ with $C_{0-}=0.025\,536\,971\,9$ \cite{barouch.1973.susceptibility}, so that $B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$ and $A=\frac\pi2\fX(0)/B^2=2^{11/8}\pi^3(\arcsinh1)^{19/4}C_{0-}$. Data was then taken for susceptibility and -magnetization for $T_\c-T,H\leq0.1$. This data is plotted in -Fig.~\ref{fig:scaling_fits}, along with collapses of data onto a single universal curve -in the insets of those figures. As can be seen, there is very good agreement +magnetization for $T_\c-T,H\leq0.1$. The resulting scaling functions $\fX$ and +$\fM$ are plotted using this data in +Fig.~\ref{fig:scaling_fits}. As can be seen, there is very good agreement between our proposed functional forms and what is measured. However, there are systematic differences that can be seen most clearly in the magnetization. Since our method is known to only be accurate for high moments of the free energy, we should expect that low moments require corrections. -Therefore, we also fit those corrections of the form +Therefore, we also fit those corrections using \begin{align} - \fX^{\twodee\prime}(X)&=\fX^\twodee(X)+\sum_{n=1}^Nf_n(BX)\\ - \fM^{\twodee\prime}(X)&=\fM^\twodee(X)+\frac{T_\c}B\sum_{n=1}^NF_n(BX) + \fX^{\twodee\prime}(X)&=\fX^\twodee(X)+\sum_{n=1}^Nf_n(BX)\label{eq:sus_scaling_poly}\\ + \fM^{\twodee\prime}(X)&=\fM^\twodee(X)+\frac{T_\c}B\sum_{n=1}^NF_n(BX)\label{eq:mag_scaling_poly} \end{align} where $F_n'(x)=f_n(x)$ and \[ \begin{aligned} f_n(x)&=\frac{C_nx^n}{1+(\lambda x)^{n+1}}\\ - F_n(x)&=\frac{C_n\lambda^{-(n+1)}}{n+1}\log(1+(\lambda x)^{n+1}) + F_n(x)&=\frac{C_n\lambda^{-(n+1)}}{n+1}\log(1+(\lambda x)^{n+1}). \end{aligned} \label{eq:poly} \] +The functions $f_n$ have been chosen to be pure integer power laws for small +argument, but vanish appropriately at large argument. We fit these functions to our numeric data for $N=3$. The resulting curves are -also plotted in Fig.~\ref{fig:scaling_fits} as a dashed line. - - - +also plotted in Fig.~\ref{fig:scaling_fits} as dashed lines. %\begin{table} % \centering @@ -357,13 +358,16 @@ also plotted in Fig.~\ref{fig:scaling_fits} as a dashed line. \begin{figure} \input{figs/fig-susmag} \caption{ - Comparisons of scaling forms \eqref{eq:sus_scaling} and - \eqref{eq:mag_scaling} to numeric data. Data with - sampling error taken from Monte Carlo simulations of a $4096\times4096$ + Scaling functions for (top) the susceptibility and (bottom) the + magnetization plotted in terms of the invarient combination + $h|t|^{-\beta\delta}$. Points with error bars show data with + sampling error taken from simulations of a $4096\times4096$ square-lattice Ising model with periodic boundary conditions and $T_\c-T=0.01,0.02,\ldots,0.1$ and - $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. The solid lines show our - analytic results, while the dashed lines have polynomial corrections of - the form \eqref{eq:poly} fit to the data for $N=3$, with $C_1=-0.00368$, + $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Color denotes the value of + $T$. The solid lines show our + analytic results \eqref{eq:sus_scaling} and \eqref{eq:mag_scaling}, while + the dashed lines show fits of \eqref{eq:sus_scaling_poly} and + \eqref{eq:mag_scaling_poly} to the data for $N=3$, with $C_1=-0.00368$, $C_2=-0.0191$, $C_3=0.0350$, and $\lambda=2.42$. } \label{fig:scaling_fits} @@ -374,7 +378,7 @@ and the analytic nature of the free energy to derive the universal scaling functions for the free energy, and in \twodee the magnetization and susceptibility, in the limit of small $t$ and $h$. Because of an essential singularity in these functions at $h=0$---the abrupt transition line---their -form cannot be modified by analytic redefinition of control or thermodynamic +form cannot be brought into that of regular functions by analytic redefinition of control or thermodynamic variables. These predictions match the results of simulations well. Having demonstrated that the essential singularity in thermodynamic functions at the abrupt singularity leads to observable effects. we hope that these functional diff --git a/figs/fig-susmag.gplot b/figs/fig-susmag.gplot index dd002a4..39a3fdf 100644 --- a/figs/fig-susmag.gplot +++ b/figs/fig-susmag.gplot @@ -50,7 +50,7 @@ set origin 0,0 set xrange [-20:1200] set yrange [1.18:1.75] set ylabel offset 1,0 '$M|t|^{-\beta}$' -set xlabel '$h|t|^{-\Delta}$' +set xlabel '$h|t|^{-\beta\delta}$' set xtics format '%g' plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A * B) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A * B + (polyint(Tc * C1, lamb, 1, 10**$1) + polyint(Tc * C2, lamb, 2, 10**$1) + polyint(Tc * C3, lamb, 3, 10**$1)) / B) smooth csplines with lines dt 2 lw 2 lc black @@ -62,7 +62,7 @@ set origin 0.31,0.5 + 0.29 / 2 set xrange [-3:3] set yrange [-4.5:-1.5] set ylabel offset 2.5,0 '\footnotesize$\chi|t|^\gamma$' -set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\Delta}$' +set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\beta\delta}$' set xtics format '\footnotesize$10^{%g}$' -2,2,3 set ytics format '\footnotesize$10^{%g}$' -4,1,-2 @@ -73,7 +73,7 @@ set origin 0.31,0.23 / 2 set xrange [-3:3] set yrange [1.18:1.75] set ylabel offset 4,0 '\footnotesize$M|t|^{-\beta}$' -set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\Delta}$' +set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\beta\delta}$' set xtics format '\footnotesize$10^{%g}$' -2,2,3 set ytics format '\footnotesize {%g}' 1.2,.2,1.6 diff --git a/makefile b/makefile index ffa4c63..b726870 100644 --- a/makefile +++ b/makefile @@ -4,13 +4,15 @@ FIGS=fig-susmag DATA=data_square-4096 FUNCS=fig-mag_scaling-func fig-sus_scaling-func -all: ${FIGS:%=figs/%.tex} - rubber $(DOC).tex - dvipdf $(DOC).dvi +all: ${DOC}.pdf figs/%.tex: figs/%.gplot ${DATA:%=data/%.dat} ${FUNCS:%=figs/%.dat} gnuplot $< > $@ +${DOC}.pdf: ${DOC}.tex ${DOC}.bib ${FIGS:%=figs/%.tex} + rubber $(DOC).tex + dvipdf $(DOC).dvi + clean: rubber --clean $(DOC) rm -f $(DOC).pdf -- cgit v1.2.3-70-g09d2 From 181dc3bdeb284c95ff68cfa29fd3cb1c0ed624a8 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Sat, 8 Jul 2017 14:09:53 -0400 Subject: make the tics logscale --- figs/fig-susmag.gplot | 33 +++++++++++++++++++-------------- 1 file changed, 19 insertions(+), 14 deletions(-) diff --git a/figs/fig-susmag.gplot b/figs/fig-susmag.gplot index 39a3fdf..da4f8e6 100644 --- a/figs/fig-susmag.gplot +++ b/figs/fig-susmag.gplot @@ -34,11 +34,13 @@ set key off set size 1,1 - 0.52 set origin 0,0.52 -set xrange [-20:1200] -set yrange [-0.5:13] +set xrange [-40:1200] +set yrange [-0.4:14.4] set ylabel offset 1,0 '$\chi|t|^{\gamma}\times 10^3$' set lmargin 6 set xtics format '' +set mxtics 5 +set mytics 5 set bmargin 0.2 plot num using (X($2, $3)):(10**3 * $10 * t($2)**gamma):(10**3 * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B):(10**(3+$2) * A * B**2 + 10**3 * (poly(C1, lamb, 1, 10**$1) + poly(C2, lamb, 2, 10**$1) + poly(C3, lamb, 3, 10**$1))) with lines dt 2 lw 2 lc black @@ -47,35 +49,38 @@ set bmargin -1 set tmargin 0.2 set size 1,0.52 set origin 0,0 -set xrange [-20:1200] -set yrange [1.18:1.75] +set yrange [1.18:1.74] set ylabel offset 1,0 '$M|t|^{-\beta}$' set xlabel '$h|t|^{-\beta\delta}$' set xtics format '%g' plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A * B) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A * B + (polyint(Tc * C1, lamb, 1, 10**$1) + polyint(Tc * C2, lamb, 2, 10**$1) + polyint(Tc * C3, lamb, 3, 10**$1)) / B) smooth csplines with lines dt 2 lw 2 lc black +set logscale xy set tmargin -1 set lmargin -1 set size 0.65,0.325 set origin 0.31,0.5 + 0.29 / 2 -set xrange [-3:3] -set yrange [-4.5:-1.5] +set xrange [0.0015:1900] +set yrange [0.00002:0.02] set ylabel offset 2.5,0 '\footnotesize$\chi|t|^\gamma$' set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\beta\delta}$' -set xtics format '\footnotesize$10^{%g}$' -2,2,3 -set ytics format '\footnotesize$10^{%g}$' -4,1,-2 +set mxtics 5 +set xtics format '' -2,10,1000 +set xtics add ('$\footnotesize10^{-2}$' 10**(-2), "" 0.1, '$\footnotesize10^0$' 1, "" 10, '$\footnotesize10^2$' 100, "" 1000) +set mytics 5 +set ytics format '\footnotesize$10^{%T}$' 0.00001,10,0.01 -plot num using (log10(X($2, $3))):(log10($10 * t($2)**gamma)):($11 / $10):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using ($1-log10(B)):($2+log10(A * B**2)) with linespoints pt 0 lw 2 lc rgb "black", susfunc using ($1-log10(B)):(log10(10**$2 * A * B**2 + poly(C1, lamb, 1, 10**$1) + poly(C2, lamb, 2, 10**$1) + poly(C3, lamb, 3, 10**$1))) with lines dt 2 lw 2 lc black +plot num using (X($2, $3)):($10 * t($2)**gamma):($11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**$2 * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B):(10**$2 * A * B**2 + poly(C1, lamb, 1, 10**$1) + poly(C2, lamb, 2, 10**$1) + poly(C3, lamb, 3, 10**$1)) with lines dt 2 lw 2 lc black +unset logscale xy +set logscale x set size 0.65,0.325 set origin 0.31,0.23 / 2 -set xrange [-3:3] set yrange [1.18:1.75] set ylabel offset 4,0 '\footnotesize$M|t|^{-\beta}$' -set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\beta\delta}$' -set xtics format '\footnotesize$10^{%g}$' -2,2,3 -set ytics format '\footnotesize {%g}' 1.2,.2,1.6 +set ytics format '\footnotesize {%g}' 1.2,0.2,1.8 +set mytics 5 -plot num using (log10(X($2, $3))):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using ($1-log10(B)):(M0 + Tc * A * B * 10**$2) with linespoints pt 0 lw 2 lc black, magfunc using ($1 - log10(B)):(M0 + 10**($2) * Tc * A * B + (polyint(Tc * C1, lamb, 1, 10**$1) + polyint(Tc * C2, lamb, 2, 10**$1) + polyint(Tc * C3, lamb, 3, 10**$1)) / B) with lines dt 2 lw 2 lc black +plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + Tc * A * B * 10**$2) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A * B + (polyint(Tc * C1, lamb, 1, 10**$1) + polyint(Tc * C2, lamb, 2, 10**$1) + polyint(Tc * C3, lamb, 3, 10**$1)) / B) with lines dt 2 lw 2 lc black -- cgit v1.2.3-70-g09d2 From 7c7e6c8459b25ea4a07e872f7373fa9b4abea010 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 11 Jul 2017 15:05:04 -0400 Subject: added log-log bars --- figs/fig-susmag.gplot | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/figs/fig-susmag.gplot b/figs/fig-susmag.gplot index da4f8e6..dfdd2a1 100644 --- a/figs/fig-susmag.gplot +++ b/figs/fig-susmag.gplot @@ -77,7 +77,7 @@ unset logscale xy set logscale x set size 0.65,0.325 set origin 0.31,0.23 / 2 -set yrange [1.18:1.75] +set yrange [1.18:1.74] set ylabel offset 4,0 '\footnotesize$M|t|^{-\beta}$' set ytics format '\footnotesize {%g}' 1.2,0.2,1.8 set mytics 5 -- cgit v1.2.3-70-g09d2 From b818025acf3c4f1973fe8548ad81e229b1f5c889 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 11 Jul 2017 16:32:27 -0400 Subject: fixed an error in the ansatz statement --- essential-ising.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/essential-ising.tex b/essential-ising.tex index a64f75b..1aa5ea1 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -214,7 +214,7 @@ ansatz that the imaginary part of the metastable free energy has the same singular behavior as the real part of the equilibrium free energy, and that for small $t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\fiF(h|t|^{-\beta\delta})$ for \[ - \fiF(X)=-A\Theta(-X)(-X)^be^{-1/(-BX)^{\dim-1}}, + \fiF(X)=-A\Theta(-X)(-BX)^be^{-1/(-BX)^{\dim-1}}, \label{eq:im.scaling} \] where $\Theta$ is the Heaviside function. Results from combining an analysis -- cgit v1.2.3-70-g09d2 From 91cf18f2f55706c7b764fdc5f048b0c6106c698e Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 12 Jul 2017 12:26:38 -0400 Subject: many final changes --- .gitignore | 1 + essential-ising.bib | 7 ++ essential-ising.tex | 288 +++++++++++++++++++++++++------------------------- figs/fig-susmag.gplot | 15 ++- makefile | 6 +- 5 files changed, 162 insertions(+), 155 deletions(-) diff --git a/.gitignore b/.gitignore index 10c9386..be37e5a 100644 --- a/.gitignore +++ b/.gitignore @@ -7,3 +7,4 @@ *.dvi figs/*.tex +fig-*.tex diff --git a/essential-ising.bib b/essential-ising.bib index 556c4e7..df4a96b 100644 --- a/essential-ising.bib +++ b/essential-ising.bib @@ -134,6 +134,13 @@ publisher={APS} } +@book{flanigan.1972.complex, + title={Complex variables: harmonic and analytic functions}, + author={Flanigan, Francis J}, + year={1972}, + publisher={Courier Corporation} +} + @article{gaveau.1989.analytic, title={Metastable decay rates and analytic continuation}, author={Gaveau, B and Schulman, LS}, diff --git a/essential-ising.tex b/essential-ising.tex index 1aa5ea1..4673847 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -76,11 +76,10 @@ used to construct a scaling ansatz for the imaginary component of the free energy of an Ising model in its metastable state close to the critical point. The analytic properties of the free energy are used to determine - scaling functions for the free energy in the vicinity of the - critical point and the abrupt transition line. These functions have - essential singularities at zero field. Analogous forms for the magnetization - and susceptibility in two-dimensions are fit to numeric data and show good - agreement. + scaling functions for the free energy in the vicinity of the critical point + and the abrupt transition line. These functions have essential singularities + at zero field. Analogous forms for the magnetization and susceptibility in + two dimensions are fit to numeric data and show good agreement. \end{abstract} \maketitle @@ -88,30 +87,32 @@ The Ising model is the canonical example of a system with a continuous phase transition, and the study of its singular properties marked the first success of the renormalization group (\textsc{rg}) method in statistical physics -\cite{wilson.1971.renormalization}. This status makes sense: it's a simple -model whose phase transition admits \textsc{rg} methods in a straightforward way, -and has exact solutions in certain dimensions and for certain parameter -restrictions. However, in one respect the Ising critical point is not simply a -continuous transition: it ends the line of abrupt phase transitions at zero -field below the critical temperature. Though typically neglected in \textsc{rg} -scaling analyses of the critical point, we demonstrate that there are -numerically measurable contributions to scaling due to the abrupt transition -line that cannot be accounted for by analytic changes of control or +\cite{wilson.1971.renormalization}. Its status makes sense: it's a simple +model whose continuous phase transition contains all the essential features of +more complex ones, but admits \textsc{rg} methods in a straightforward way and +has exact solutions in certain dimensions and for certain parameter +restrictions. However, the Ising critical point is not simply a continuous +transition: it also ends a line of abrupt phase transitions extending from it +at zero field below the critical temperature. Though typically neglected in +\textsc{rg} scaling analyses of the critical point, we demonstrate that there +are numerically measurable contributions to scaling due to the abrupt +transition line that cannot be accounted for by analytic changes of control or thermodynamic variables. \textsc{Rg} analysis predicts that the singular part of the free energy per -site $F$ as a function of reduced temperature $t=1-T_\c/T$ and field $h=H/T$ in -the vicinity of the critical point takes the scaling form -$F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})$ for the low temperature phase $t<0$ \cite{cardy.1996.scaling,aharony.1983.fields}. When studying the properties of -the Ising critical point, it is nearly always assumed that $\fF(X)$, -the universal scaling function, is an analytic function of $X$. However, it -has long been known that there exists an essential singularity in $\fF$ -at $X=0$, though its effects have long been believed to be unobservable +site $F$ as a function of reduced temperature $t=1-T_\c/T$ and field $h=H/T$ +in the vicinity of the critical point takes the scaling form +$F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})$ for the low temperature phase +$t<0$ \cite{cardy.1996.scaling}. When studying the properties of the Ising +critical point, it is nearly always assumed that the universal scaling +function $\fF$ is analytic, i.e., has a convergent Taylor series. However, it +has long been known that there exists an essential singularity in $\fF$ at +zero argument, though its effects have long been believed to be unobservable \cite{fisher.1967.condensation}, or simply just neglected \cite{guida.1997.3dising,schofield.1969.parametric,schofield.1969.correlation,caselle.2001.critical,josephson.1969.equation,fisher.1999.trigonometric}. With careful analysis, we have found that assuming the presence of the essential singularity is predictive of the scaling form of e.g. the -susceptibility. +susceptibility and magnetization. The provenance of the essential singularity can be understood using the methods of critical droplet theory for the decay of an Ising system in a @@ -136,19 +137,21 @@ In critical droplet theory, the metastable state decays when a domain of the equilibrium state forms whose surface-energy cost for growth is outweighed by bulk-energy gains. There is numerical evidence that, near the critical point, droplets are spherical \cite{gunther.1993.transfer-matrix}. The free energy -cost of the surface of a droplet of radius $R$ -is $\Sigma S_\dim R^{\dim-1}$ and that of its -bulk is $-M|H|V_\dim R^\dim$, where $S_\dim$ and $V_\dim$ are the surface area and -volume of a $(\dim-1)$-sphere, respectively, and $\Sigma$ is the surface tension of the equilibrium--metastable interface. The critical droplet size then is -$R_\c=(\dim-1)\Sigma/M|H|$ and the free energy of the critical -droplet is $\Delta -F_\c=\pi^{\dim/2}\Sigma^\dim((\dim-1)/M|H|)^{\dim-1}/\Gamma(1+\dim/2)$. +cost of the surface of a droplet of radius $R$ is $\Sigma S_\dim R^{\dim-1}$ +and that of its bulk is $-M|H|V_\dim R^\dim$, where $S_\dim$ and $V_\dim$ are +the surface area and volume of a $(\dim-1)$-sphere, respectively, and $\Sigma$ +is the surface tension of the equilibrium--metastable interface. The critical +droplet size then is $R_\c=(\dim-1)\Sigma/M|H|$ and the free energy of the +critical droplet is $\Delta +F_\c=\pi^{\dim/2}\Sigma^\dim((\dim-1)/M|H|)^{\dim-1}/\Gamma(1+\dim/2)$. Assuming the typical singular scaling forms $\Sigma/T=|t|^\mu\fS(h|t|^{-\beta\delta})$ and $M=|t|^\beta\mathcal M(h|t|^{-\beta\delta})$ and using known hyperscaling relations \cite{widom.1981.interface}, this implies a scaling form \def\eqcritformone{ - T\frac{\pi^{\dim/2}(\dim-1)^{\dim-1}}{\Gamma(1+\dim/2)}\frac{\fS^\dim(h|t|^{-\beta\delta})}{(-h|t|^{-\beta\delta}\fM(h|t|^{-\beta\delta}))^{\dim-1}} + T\frac{\pi^{\dim/2}(\dim-1)^{\dim-1}}{\Gamma(1+\dim/2)} + \frac{\fS^\dim(h|t|^{-\beta\delta})}{(-h|t|^{-\beta\delta} + \fM(h|t|^{-\beta\delta}))^{\dim-1}} } \def\eqcritformtwo{ T\fG^{-(\dim-1)}(h|t|^{-\beta\delta}) @@ -169,70 +172,55 @@ M(h|t|^{-\beta\delta})$ and using known hyperscaling relations Since both surface tension and magnetization are finite and nonzero for $H=0$ at $T1$ this function has an essential singularity in the invariant combination $h|t|^{-\beta\delta}$. -% $\Gamma/\Gamma_\sq=(D_\sq/D)(\fM(0)^{1/\beta}/\fM_\sq(0)^{1/\beta})^{-7/4}$ - This form of $\im F$ for small $h$ is well known -\cite{langer.1967.condensation,harris.1984.metastability}. We make the scaling -ansatz that the imaginary part of the metastable free energy has the same -singular behavior as the real part of the equilibrium free energy, and that for -small $t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\fiF(h|t|^{-\beta\delta})$ for +\cite{langer.1967.condensation,harris.1984.metastability}. We make the +scaling ansatz that the imaginary part of the metastable free energy has the +same singular behavior as the real part of the equilibrium free energy, and +that for small $t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\fiF(h|t|^{-\beta\delta})$, +where \[ - \fiF(X)=-A\Theta(-X)(-BX)^be^{-1/(-BX)^{\dim-1}}, + \fiF(X)=-A\Theta(-X)(-BX)^be^{-1/(-BX)^{\dim-1}} \label{eq:im.scaling} \] -where $\Theta$ is the Heaviside function. Results from combining an analysis -of fluctuations on the surface of critical droplets with \textsc{rg} recursion -relations suggest that $b=-(d-3)d/2$ for $d=2,4$ and $b=-7/3$ for -$d=3$ +and $\Theta$ is the Heaviside function. Results from combining an analysis of +fluctuations on the surface of critical droplets with \textsc{rg} recursion +relations suggest that $b=-(d-3)d/2$ for $d=2,4$ and $b=-7/3$ for $d=3$ \cite{houghton.1980.metastable,rudnick.1976.equations,gunther.1980.goldstone}. Assuming that $F$ is analytic in the upper complex-$h$ plane, the real part of $F$ in the equilibrium state can be extracted from this imaginary metastable free energy using the Kramers--Kronig relation \[ \re F(t,h)=\frac1\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{h'-h}\,\dd h'. + \label{eq:kram-kron} \] This relationship has been used to compute high-order moments of the free energy in $H$ in good agreement with transfer matrix expansions \cite{lowe.1980.instantons}. Here, we compute the integral to come to explicit -functional forms. In \threedee and \fourdee this can be computed -explicitly given our scaling ansatz, yielding +functional forms. In \threedee and \fourdee this can be computed explicitly +given our scaling ansatz, yielding \def\eqthreedeeone{ \fF^\threedee(Y/B)&= \frac{A}{12}\frac{e^{-1/Y^2}}{Y^2} @@ -254,14 +242,14 @@ explicitly given our scaling ansatz, yielding \begin{align} &\begin{aligned} \eqthreedeeone\\ - &\hspace{6em} + &\hspace{8em} \eqthreedeetwo \end{aligned} \\ &\begin{aligned} \eqfourdeeone \\ - &\hspace{2em} + &\hspace{-0.5em} \eqfourdeetwo. \end{aligned} \end{align} @@ -272,119 +260,131 @@ explicitly given our scaling ansatz, yielding \eqfourdeeone\eqfourdeetwo. \end{align} \fi -At the level of truncation we are working at, the Kramers--Kronig relation -does not converge in \twodee. However, the higher moments can still be -extracted, e.g., the susceptibility, by taking +At the level of truncation of \eqref{eq:im.scaling} at which we are working +the Kramers--Kronig relation does not converge in \twodee. However, higher +moments can still be extracted, e.g., the susceptibility, by taking \[ \chi=\pd[2]Fh =\frac2\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. \] -With $\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, this yields +With a scaling form defined by $\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, +this yields \[ \fX^\twodee(Y/B)=\frac{AB^2}{\pi Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big] \label{eq:sus_scaling} \] -The constant $A$ can be fixed by zero-field results, with $\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=\frac{2AB^2}\pi$. Scaling forms for the free energy can then be extracted by direct integration and their constants of integration fixed by known zero field values, yielding \begin{align} \label{eq:mag_scaling} \fM^\twodee(Y/B) - &=\fM(0)+\frac{ABT_c}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\ + &=\fM(0)+\frac{ABT_\c}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\ \fF^\twodee(Y/B) - &=\fF(0)+T_cY\bigg(\frac{\fM(0)}B+\frac{AT_c}\pi e^{1/Y}\ei(-1/Y)\bigg) + &=\fF(0)+T_\c Y\bigg(\frac{\fM(0)}B+\frac{AT_\c}\pi e^{1/Y}\ei(-1/Y)\bigg) \end{align} -with $F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})+t^{2-\alpha}\log|t|$ in two dimensions. +with $F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})+t^{2-\alpha}\log|t|$ in +two dimensions. -How are these functional forms to be interpreted? They are not asymptotic -forms in any sense, as there is no limit of $t$ or $h$ in which they become -aribitrarily correct. It is well established that this method of using +How are these functional forms to be interpreted? Though the scaling function +\eqref{eq:im.scaling} for the imaginary free energy of the metastable state is +asymptotically correct sufficiently close to the critical point, the results +of the integral relation \eqref{eq:kram-kron} are not, since there is no limit +of $t$ or $h$ in which it becomes arbitrarily correct for a given truncation +of \eqref{eq:im.scaling}. It is well established that this method of using unphysical or metastable elements of a theory to extract properties of the -stable or equilibrium theory only works for high moments of those predictions -\cite{parisi.1977.asymptotic,bogomolny.1977.dispersion}. -These functions should be understood as possessing exactly the correct +stable or equilibrium theory is only accurate for high moments of those +predictions \cite{parisi.1977.asymptotic,bogomolny.1977.dispersion}. The +functions above should be understood as possessing exactly the correct singularity at the coexistence line, but requiring polynomial corrections, especially for smaller integer powers. Using these forms in conjunction with existing methods of describing the critical equation of state or critical properties with analytic functions in $h$ will incorporate these low-order -corrections while preserving the correct singular structure. +corrections while preserving the correct singular structure. In other words, +the scaling functions can be \emph{exactly} described by +$\tilde\fF(X)=\fF(X)+f(X)$ for some analytic function $f$. Higher order terms +in the expansion of $\tilde\fF$ become asymptotically equal to those of $\fF$ +because, as an analytic function, progressively higher order terms of $f$ must +eventually become arbitrarily small \cite{flanigan.1972.complex}. How predictive are these scaling forms in the proximity of the critical point -and the abrupt transition line? We simulated the \twodee Ising model on square lattice using a form of the Wolff algorithm modified -to remain efficient in the presence of an external field. Briefly, the external field $H$ is applied by adding an extra spin $s_0$ with coupling $|H|$ to all others -\cite{dimitrovic.1991.finite}. A quickly converging estimate for the magnetization in the finite-size system was then made by taking $M=\sgn(H)s_0\sum s_i$, i.e., the magnetization relative to the external spin. For the \twodee Ising model on a square lattice, exact results at zero temperature have $\fS(0)=4/T_c$, $\fM(0)=(2^{5/2}\arcsinh1)^\beta$ \cite{onsager.1944.crystal}, and $\fX(0)=C_{0-}/T_\c$ with $C_{0-}=0.025\,536\,971\,9$ -\cite{barouch.1973.susceptibility}, so that $B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$ and $A=\frac\pi2\fX(0)/B^2=2^{11/8}\pi^3(\arcsinh1)^{19/4}C_{0-}$. -Data was then taken for susceptibility and -magnetization for $T_\c-T,H\leq0.1$. The resulting scaling functions $\fX$ and -$\fM$ are plotted using this data in +and the abrupt transition line? We simulated the \twodee Ising model on square +lattice using a form of the Wolff algorithm modified to remain efficient in +the presence of an external field. Briefly, the external field $H$ is applied +by adding an extra spin $s_0$ with coupling $|H|$ to all others +\cite{dimitrovic.1991.finite}. A quickly converging estimate for the +magnetization in the finite-size system was then made by taking +$M=\sgn(H)s_0\sum s_i$, i.e., the magnetization relative to the external spin. +Data was then taken for susceptibility and magnetization for +$T_\c-T,H\leq0.1$. This data, rescaled as appropriate to collapse onto a +single curve, is plotted in Fig.~\ref{fig:scaling_fits}. + +For the \twodee Ising model on a square lattice, exact results at zero +temperature have $\fS(0)=4/T_\c$, $\fM(0)=(2^{5/2}\arcsinh1)^\beta$ +\cite{onsager.1944.crystal}, and $\fX(0)=C_0^-/T_\c$ with +$C_0^-=0.025\,536\,971\,9$ \cite{barouch.1973.susceptibility}, so that +$B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$. If we +assume incorrectly that \eqref{eq:sus_scaling} is the true asymptotic form of +the susceptibility scaling function, then +$\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ and the constant +$A$ is fixed to $A=\pi\fX(0)/2B^2=2^{11/8}\pi^3(\arcsinh1)^{19/4}C_0^-$. The +resulting scaling functions $\fX$ and $\fM$ are plotted as solid lines in Fig.~\ref{fig:scaling_fits}. As can be seen, there is very good agreement -between our proposed functional forms and what is measured. -However, there are systematic differences that can be seen most clearly in the -magnetization. Since our method is known to only be accurate for high moments -of the free energy, we should expect that low moments require corrections. -Therefore, we also fit those corrections using +between our proposed functional forms and what is measured. However, there +are systematic differences that can be seen most clearly in the magnetization. +Since our method is known to only be accurate for high moments of the free +energy, we should expect that low moments require corrections. Therefore, we +also fit those corrections using \begin{align} - \fX^{\twodee\prime}(X)&=\fX^\twodee(X)+\sum_{n=1}^Nf_n(BX)\label{eq:sus_scaling_poly}\\ - \fM^{\twodee\prime}(X)&=\fM^\twodee(X)+\frac{T_\c}B\sum_{n=1}^NF_n(BX)\label{eq:mag_scaling_poly} + \tilde\fX^\twodee(X)&=\fX^\twodee(X)+\sum_{n=0}^Nf_n(BX)\label{eq:sus_scaling_poly}\\ + \tilde\fM^\twodee(X)&=\fM^\twodee(X)+\frac{T_\c}B\sum_{n=0}^NF_n(BX)\label{eq:mag_scaling_poly} \end{align} -where $F_n'(x)=f_n(x)$ and +where $F_n'(Y)=f_n(Y)$ and \[ \begin{aligned} - f_n(x)&=\frac{C_nx^n}{1+(\lambda x)^{n+1}}\\ - F_n(x)&=\frac{C_n\lambda^{-(n+1)}}{n+1}\log(1+(\lambda x)^{n+1}). + f_n(Y)&=\frac{c_nx^n}{1+(\lambda Y)^{n+1}}\\ + F_n(Y)&=\frac{c_n\lambda^{-(n+1)}}{n+1}\log(1+(\lambda Y)^{n+1}). \end{aligned} \label{eq:poly} \] The functions $f_n$ have been chosen to be pure integer power laws for small -argument, but vanish appropriately at large argument. -We fit these functions to our numeric data for $N=3$. The resulting curves are -also plotted in Fig.~\ref{fig:scaling_fits} as dashed lines. - -%\begin{table} -% \centering -% \begin{tabular}{c|llc} -% Lattice & $T_\c$ & $\fM(0)^{1/\beta}$ & $D/D_\sq$ \\ -% \hline % ------------------------------------------------------------------- -% Square & $2/\log(1+\sqrt2)$ & $2^{5/2}\arcsinh1$ & 1 \\ -% Triangular & $4/\log3$ & $4\log3$ & $3^{3/2}/4$ \\ -% Hexagonal & $2/\log(2+\sqrt3)$ & $\frac8{\sqrt3}\arccosh2$ & $3^{3/2}/8$ -% \end{tabular} -% \caption{ -% The critical temperatures and amplitudes for the magetization along both the coexistence line and the critical isotherm, for three different lattices. -% } -% \label{tab:consts} -%\end{table} +argument, but vanish appropriately at large argument. This is necessary +because the susceptibility vanishes with $h|t|^{-\beta\delta}$ while bare +polynomial corrections would not. We fit these functions to our numeric data +for $N=0$ while requiring that $C_0^-/T_\c=\fX'(0)=c_0+2AB^2/\pi$. The +resulting curves are also plotted as dashed lines in +Fig.~\ref{fig:scaling_fits}. Our singular scaling function with one low-order +correction appears to match data quite well. \begin{figure} - \input{figs/fig-susmag} + \input{fig-susmag} \caption{ Scaling functions for (top) the susceptibility and (bottom) the - magnetization plotted in terms of the invarient combination - $h|t|^{-\beta\delta}$. Points with error bars show data with - sampling error taken from simulations of a $4096\times4096$ - square-lattice Ising model with periodic boundary conditions and $T_\c-T=0.01,0.02,\ldots,0.1$ and - $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Color denotes the value of - $T$. The solid lines show our - analytic results \eqref{eq:sus_scaling} and \eqref{eq:mag_scaling}, while - the dashed lines show fits of \eqref{eq:sus_scaling_poly} and - \eqref{eq:mag_scaling_poly} to the data for $N=3$, with $C_1=-0.00368$, - $C_2=-0.0191$, $C_3=0.0350$, and $\lambda=2.42$. + magnetization plotted in terms of the invariant combination + $h|t|^{-\beta\delta}$. Points with error bars show data with sampling + error taken from simulations of a $4096\times4096$ square-lattice Ising + model with periodic boundary conditions and $T_\c-T=0.01,0.02,\ldots,0.1$ + and $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Color denotes the value of + $T$, with red corresponding to $0.01$ and violet to $0.1$. The solid lines + show our analytic results \eqref{eq:sus_scaling} and + \eqref{eq:mag_scaling}, while the dashed lines show fits of + \eqref{eq:sus_scaling_poly} and \eqref{eq:mag_scaling_poly} to the data + for $N=0$, with $c_0=-0.0124$ and $\lambda=1.77$. } \label{fig:scaling_fits} \end{figure} We have used results from the properties of the metastable Ising ferromagnet -and the analytic nature of the free energy to derive the universal scaling +and the analytic nature of the free energy to derive universal scaling functions for the free energy, and in \twodee the magnetization and -susceptibility, in the limit of small $t$ and $h$. Because of an essential +susceptibility, in the limit of small $t<0$ and $h$. Because of an essential singularity in these functions at $h=0$---the abrupt transition line---their -form cannot be brought into that of regular functions by analytic redefinition of control or thermodynamic -variables. These predictions match the results of simulations well. Having -demonstrated that the essential singularity in thermodynamic functions at the -abrupt singularity leads to observable effects. we hope that these functional -forms will be used in conjunction with traditional perturbation methods to -better express the equation of state of the Ising model in the whole of its -parameter space. +form cannot be brought into that of any regular function by analytic +redefinition of control or thermodynamic variables. These predictions match +the results of simulations well. Having demonstrated that the essential +singularity in thermodynamic functions at the abrupt transition leads to +observable scaling effects, we hope that these functional forms will be used in +conjunction with traditional perturbation methods to better express the +equation of state of the Ising model in the whole of its parameter space. \begin{acknowledgments} The authors would like to thank Tom Lubensky, Andrea Liu, and Randy Kamien diff --git a/figs/fig-susmag.gplot b/figs/fig-susmag.gplot index dfdd2a1..9682953 100644 --- a/figs/fig-susmag.gplot +++ b/figs/fig-susmag.gplot @@ -19,10 +19,9 @@ B = Tc**2 * M0 / (16 * pi) C0 = 0.0255369719 A = pi / 2 * C0 / (B**2 * Tc) -C1 = -0.00368218 -C2 = -0.0190753 -C3 = 0.0349601 -lamb = 2.41718 +c0 = -0.012384 +lamb = 1.76962 +A2 = pi / 2 * (C0 / Tc - c0) / B**2 susfunc = "figs/fig-sus_scaling-func.dat" magfunc = "figs/fig-mag_scaling-func.dat" @@ -43,7 +42,7 @@ set mxtics 5 set mytics 5 set bmargin 0.2 -plot num using (X($2, $3)):(10**3 * $10 * t($2)**gamma):(10**3 * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B):(10**(3+$2) * A * B**2 + 10**3 * (poly(C1, lamb, 1, 10**$1) + poly(C2, lamb, 2, 10**$1) + poly(C3, lamb, 3, 10**$1))) with lines dt 2 lw 2 lc black +plot num using (X($2, $3)):(10**3 * $10 * t($2)**gamma):(10**3 * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B):(10**(3+$2) * A2 * B**2 + 10**3 * poly(c0, lamb, 0, 10**$1)) with lines dt 2 lw 2 lc black set bmargin -1 set tmargin 0.2 @@ -54,7 +53,7 @@ set ylabel offset 1,0 '$M|t|^{-\beta}$' set xlabel '$h|t|^{-\beta\delta}$' set xtics format '%g' -plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A * B) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A * B + (polyint(Tc * C1, lamb, 1, 10**$1) + polyint(Tc * C2, lamb, 2, 10**$1) + polyint(Tc * C3, lamb, 3, 10**$1)) / B) smooth csplines with lines dt 2 lw 2 lc black +plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A * B) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A2 * B + polyint(Tc * c0, lamb, 0, 10**$1) / B) smooth csplines with lines dt 2 lw 2 lc black set logscale xy set tmargin -1 @@ -71,7 +70,7 @@ set xtics add ('$\footnotesize10^{-2}$' 10**(-2), "" 0.1, '$\footnotesize10^0$' set mytics 5 set ytics format '\footnotesize$10^{%T}$' 0.00001,10,0.01 -plot num using (X($2, $3)):($10 * t($2)**gamma):($11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**$2 * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B):(10**$2 * A * B**2 + poly(C1, lamb, 1, 10**$1) + poly(C2, lamb, 2, 10**$1) + poly(C3, lamb, 3, 10**$1)) with lines dt 2 lw 2 lc black +plot num using (X($2, $3)):($10 * t($2)**gamma):($11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**$2 * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B):(10**$2 * A2 * B**2 + poly(c0, lamb, 0, 10**$1)) with lines dt 2 lw 2 lc black unset logscale xy set logscale x @@ -82,5 +81,5 @@ set ylabel offset 4,0 '\footnotesize$M|t|^{-\beta}$' set ytics format '\footnotesize {%g}' 1.2,0.2,1.8 set mytics 5 -plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + Tc * A * B * 10**$2) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A * B + (polyint(Tc * C1, lamb, 1, 10**$1) + polyint(Tc * C2, lamb, 2, 10**$1) + polyint(Tc * C3, lamb, 3, 10**$1)) / B) with lines dt 2 lw 2 lc black +plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + Tc * A * B * 10**$2) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A2 * B + polyint(Tc * c0, lamb, 0, 10**$1) / B) with lines dt 2 lw 2 lc black diff --git a/makefile b/makefile index b726870..32f4b74 100644 --- a/makefile +++ b/makefile @@ -6,10 +6,10 @@ FUNCS=fig-mag_scaling-func fig-sus_scaling-func all: ${DOC}.pdf -figs/%.tex: figs/%.gplot ${DATA:%=data/%.dat} ${FUNCS:%=figs/%.dat} +%.tex: figs/%.gplot ${DATA:%=data/%.dat} ${FUNCS:%=figs/%.dat} gnuplot $< > $@ -${DOC}.pdf: ${DOC}.tex ${DOC}.bib ${FIGS:%=figs/%.tex} +${DOC}.pdf: ${DOC}.tex ${DOC}.bib ${FIGS:%=%.tex} rubber $(DOC).tex dvipdf $(DOC).dvi @@ -17,5 +17,5 @@ clean: rubber --clean $(DOC) rm -f $(DOC).pdf rm -f $(DOC)Notes.bib - rm -f ${FIGS:%=figs/%.tex} + rm -f ${FIGS:%=%.tex} -- cgit v1.2.3-70-g09d2 From 9223613cbd237b833adc256550b1e8e0e4dead0b Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 26 Jul 2017 13:20:34 -0400 Subject: added prl statement, shout-out at Langer for being so cool --- essential-ising.tex | 3 ++- essential-ising_prl.txt | 18 ++++++++++++++++++ 2 files changed, 20 insertions(+), 1 deletion(-) create mode 100644 essential-ising_prl.txt diff --git a/essential-ising.tex b/essential-ising.tex index 4673847..0c42b65 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -388,7 +388,8 @@ equation of state of the Ising model in the whole of its parameter space. \begin{acknowledgments} The authors would like to thank Tom Lubensky, Andrea Liu, and Randy Kamien - for helpful conversations. This work was partially supported by NSF grant + for helpful conversations. We would also like to thank Jim Langer for his + insightful canonical papers on this subject. This work was partially supported by NSF grant DMR-1312160. \end{acknowledgments} diff --git a/essential-ising_prl.txt b/essential-ising_prl.txt new file mode 100644 index 0000000..41039c1 --- /dev/null +++ b/essential-ising_prl.txt @@ -0,0 +1,18 @@ + +The renormalization group and the scaling theories that result from its use +have become essential to our understanding of singular behavior in +thermodynamic functions at continuous phase transitions. Despite this, they +have long been viewed as irrelevant to the quantitative study of abrupt phase +transitions, and likewise the presence of an abrupt phase transition has +typically been considered irrelevant to scaling analysis. Our work +demonstrates, for one common model with both a continuous and a +line of abrupt transitions, that well-known singularities in thermodynamic +functions at the abrupt transition long assumed to be unobservable are in fact +part of the scaling theory of the critical point, and make explicit +predictions confirmed by numerics. This result indicates that many published +statistical models whose abrupt transitions were not incorporated into scaling +theories should be revisited, and introduces a novel new method for +determining new universal scaling properties in previously poorly-understood +regimes. + + -- cgit v1.2.3-70-g09d2 From 5a2365cecda885db2278ba8e9621b96899f72b21 Mon Sep 17 00:00:00 2001 From: "James P. Sethna" Date: Thu, 27 Jul 2017 12:18:50 -0400 Subject: JPS edits to paper and ackn --- essential-ising.tex | 49 ++++++++++++++++++++++++++++++++++--------------- 1 file changed, 34 insertions(+), 15 deletions(-) diff --git a/essential-ising.tex b/essential-ising.tex index 0c42b65..cdd7cd6 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -373,24 +373,43 @@ correction appears to match data quite well. \label{fig:scaling_fits} \end{figure} -We have used results from the properties of the metastable Ising ferromagnet -and the analytic nature of the free energy to derive universal scaling -functions for the free energy, and in \twodee the magnetization and -susceptibility, in the limit of small $t<0$ and $h$. Because of an essential -singularity in these functions at $h=0$---the abrupt transition line---their -form cannot be brought into that of any regular function by analytic -redefinition of control or thermodynamic variables. These predictions match -the results of simulations well. Having demonstrated that the essential -singularity in thermodynamic functions at the abrupt transition leads to -observable scaling effects, we hope that these functional forms will be used in -conjunction with traditional perturbation methods to better express the -equation of state of the Ising model in the whole of its parameter space. +Abrupt phase transitions, such as the jump in magnetization in the Ising +model below $T_c$, are known to imply essential singularities in the free +energy that are usually thought to be unobservable in practice. We have +argued that this essential singularity controls the universal scaling +behavior near continuous phase transitions, and have derived an explicit +analytical form for the singularity in the free energy, magnetization, +and susceptibility for the Ising model. We have developed a Wolff algorithm +for the Ising model in a field, and showed that incorporating our singularity +into the scaling function gives good convergence to the simulations in \twodee. + +Our results should allow improved high-precision functional forms for the free +energy~\cite{CaselleXXX}, and should have implications for the scaling +of correlation functions~\cite{YJXXX,XXX}. Our methods might be generalized +to predict similar singularities in systems where nucleation and metastability +are proximate to continuous phase transitions, such as 2D superfluid +transitions~\cite{ALHN}, the melting of 2D crystals~\cite{XXX}, and +freezing transitions in glasses, spin glasses, and other disordered systems. + + +%We have used results from the properties of the metastable Ising ferromagnet +%and the analytic nature of the free energy to derive universal scaling +%functions for the free energy, and in \twodee the magnetization and +%susceptibility, in the limit of small $t<0$ and $h$. Because of an essential +%singularity in these functions at $h=0$---the abrupt transition line---their +%form cannot be brought into that of any regular function by analytic +%redefinition of control or thermodynamic variables. These predictions match +%the results of simulations well. Having demonstrated that the essential +%singularity in thermodynamic functions at the abrupt transition leads to +%observable scaling effects, we hope that these functional forms will be used in +%conjunction with traditional perturbation methods to better express the +%equation of state of the Ising model in the whole of its parameter space. \begin{acknowledgments} The authors would like to thank Tom Lubensky, Andrea Liu, and Randy Kamien - for helpful conversations. We would also like to thank Jim Langer for his - insightful canonical papers on this subject. This work was partially supported by NSF grant - DMR-1312160. + for helpful conversations. JPS thanks Jim Langer for past inspiration, + guidance, and encouragement. This work was supported by NSF grants + DMR-1312160 and DMR-1719490. \end{acknowledgments} \bibliography{essential-ising} -- cgit v1.2.3-70-g09d2 From f619693ba6cb25112c92af4dc555d8a1499ac32c Mon Sep 17 00:00:00 2001 From: "James P. Sethna" Date: Thu, 27 Jul 2017 12:53:23 -0400 Subject: New draft for 100 word blurb --- .essential-ising.tex.swp | Bin 0 -> 16384 bytes essential-ising_prl2.txt | 1 + 2 files changed, 1 insertion(+) create mode 100644 .essential-ising.tex.swp create mode 100644 essential-ising_prl2.txt diff --git a/.essential-ising.tex.swp b/.essential-ising.tex.swp new file mode 100644 index 0000000..e091a93 Binary files /dev/null and b/.essential-ising.tex.swp differ diff --git a/essential-ising_prl2.txt b/essential-ising_prl2.txt new file mode 100644 index 0000000..1cc1f44 --- /dev/null +++ b/essential-ising_prl2.txt @@ -0,0 +1 @@ +The Ising model critical point governs the termination of a line of abrupt transitions (e.g., the liquid-gas transition endpoint at high pressure). The universal scaling functions describing the critical behavior in non-zero field have long been a challenge. We show that these scaling functions inherit an essential singularity from the abrupt transition line, associated with the nucleation of droplets in the metastable phase (e.g. the supercooled gas), giving an analytic formula for the singularity and fitting to simulations. This same basic method may be useful in studying other critical points with metastability, such as in glasses and disordered systems. -- cgit v1.2.3-70-g09d2 From e45540b621cf266420f98c0d26ce60378aad4652 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Thu, 27 Jul 2017 13:24:44 -0400 Subject: made some small merge changes --- .essential-ising.tex.swp | Bin 16384 -> 0 bytes essential-ising.tex | 4 ++-- 2 files changed, 2 insertions(+), 2 deletions(-) delete mode 100644 .essential-ising.tex.swp diff --git a/.essential-ising.tex.swp b/.essential-ising.tex.swp deleted file mode 100644 index e091a93..0000000 Binary files a/.essential-ising.tex.swp and /dev/null differ diff --git a/essential-ising.tex b/essential-ising.tex index cdd7cd6..ea8e9eb 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -374,7 +374,7 @@ correction appears to match data quite well. \end{figure} Abrupt phase transitions, such as the jump in magnetization in the Ising -model below $T_c$, are known to imply essential singularities in the free +model below $T_\c$, are known to imply essential singularities in the free energy that are usually thought to be unobservable in practice. We have argued that this essential singularity controls the universal scaling behavior near continuous phase transitions, and have derived an explicit @@ -384,7 +384,7 @@ for the Ising model in a field, and showed that incorporating our singularity into the scaling function gives good convergence to the simulations in \twodee. Our results should allow improved high-precision functional forms for the free -energy~\cite{CaselleXXX}, and should have implications for the scaling +energy~\cite{caselle.2001.critical}, and should have implications for the scaling of correlation functions~\cite{YJXXX,XXX}. Our methods might be generalized to predict similar singularities in systems where nucleation and metastability are proximate to continuous phase transitions, such as 2D superfluid -- cgit v1.2.3-70-g09d2 From 0d76bc2985132cdc8199fb542ee9ef6178a0c01d Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 2 Aug 2017 22:47:40 -0400 Subject: many changes to incorporate new low-order info, small corrections to scaling functions --- essential-ising.bib | 61 +++++++++++++++++++++++++++++++++++++++++++ essential-ising.tex | 34 ++++++++++++------------ figs/fig-susmag.gplot | 71 ++++++++++++++++++++++++++++++++++++++++++++++----- 3 files changed, 143 insertions(+), 23 deletions(-) diff --git a/essential-ising.bib b/essential-ising.bib index df4a96b..16a0883 100644 --- a/essential-ising.bib +++ b/essential-ising.bib @@ -1,4 +1,22 @@ +@article{ambegaokar.1978.dissipation, + author = "V. Ambegaokar and B. I. Halperin and D. Nelson and E. Siggia", + year = "1978", + journal = "Physical Review Letters", + volume = "40", + pages = "783--6", + title = "Dissipation in two-dimensional superfluids", +} + +@article{ambegaokar.1980.dynamics, + author = "V. Ambegaokar and B. I. Halperin and D. Nelson and E. Siggia", + year = "1980", + journal = "Physical Review B", + volume = "21", + pages = "1806--26", + title = "Dynamics of superfluid films", +} + @article{aharony.1983.fields, title={Nonlinear scaling fields and corrections to scaling near criticality}, author={Aharony, Amnon and Fisher, Michael E}, @@ -90,6 +108,14 @@ publisher={IOP Publishing} } +@article{chen.2013.universal, + title={Universal scaling function for the two-dimensional Ising model in an external field: A pragmatic approach}, + author={Chen, Yan-Jiun and Paquette, Natalie M and Machta, Benjamin B and Sethna, James P}, + journal={arXiv preprint arXiv:1307.6899}, + year={2013} +} + + @article{dimitrovic.1991.finite, title={Finite-size effects, goldstone bosons and critical exponents in the d= 3 Heisenberg model}, author={Dimitrovi{\'c}, I and Hasenfratz, P and Nager, J and Niedermayer, Ferenc}, @@ -294,6 +320,28 @@ publisher={IOP Publishing} } +@article{mangazeev.2008.variational, + title={Variational approach to the scaling function of the 2D Ising model in a magnetic field}, + author={Mangazeev, Vladimir V and Batchelor, Murray T and Bazhanov, Vladimir V and Dudalev, Michael Yu}, + journal={Journal of Physics A: Mathematical and Theoretical}, + volume={42}, + number={4}, + pages={042005}, + year={2008}, + publisher={IOP Publishing} +} + +@article{mangazeev.2010.scaling, + title={Scaling and universality in the two-dimensional Ising model with a magnetic field}, + author={Mangazeev, Vladimir V and Dudalev, Michael Yu and Bazhanov, Vladimir V and Batchelor, Murray T}, + journal={Physical Review E}, + volume={81}, + number={6}, + pages={060103}, + year={2010}, + publisher={APS} +} + @article{mccraw.1978.metastability, title={Metastability in the two-dimensional Ising model}, author={McCraw, RJ and Schulman, LS}, @@ -413,6 +461,18 @@ publisher={APS} } +@article{wu.1976.spin, + title={Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region}, + author={Wu, Tai Tsun and McCoy, Barry M and Tracy, Craig A and Barouch, Eytan}, + journal={Physical Review B}, + volume={13}, + number={1}, + pages={316}, + year={1976}, + publisher={APS} +} + + @article{zinn.1996.universal, title={Universal surface-tension and critical-isotherm amplitude ratios in three dimensions}, author={Zinn, Shun-yong and Fisher, Michael E}, @@ -424,3 +484,4 @@ publisher={Elsevier} } + diff --git a/essential-ising.tex b/essential-ising.tex index ea8e9eb..c98e938 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -202,7 +202,7 @@ same singular behavior as the real part of the equilibrium free energy, and that for small $t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\fiF(h|t|^{-\beta\delta})$, where \[ - \fiF(X)=-A\Theta(-X)(-BX)^be^{-1/(-BX)^{\dim-1}} + \fiF(X)=A\Theta(-X)(-BX)^be^{-1/(-BX)^{\dim-1}} \label{eq:im.scaling} \] and $\Theta$ is the Heaviside function. Results from combining an analysis of @@ -224,14 +224,14 @@ given our scaling ansatz, yielding \def\eqthreedeeone{ \fF^\threedee(Y/B)&= \frac{A}{12}\frac{e^{-1/Y^2}}{Y^2} - \bigg[4Y\Gamma(\tfrac23)E_{5/3}(-Y^{-2}) + \bigg[\Gamma(\tfrac16)E_{7/6}(-Y^{-2}) } \def\eqthreedeetwo{ - -\Gamma(\tfrac16)E_{7/6}(-Y^{-2})\bigg] + -4Y\Gamma(\tfrac23)E_{5/3}(-Y^{-2})\bigg] } \def\eqfourdeeone{ \fF^\fourdee(Y/B)&= - \frac{A}{9\pi}\frac{e^{1/Y^3}}{Y^2} + -\frac{A}{9\pi}\frac{e^{1/Y^3}}{Y^2} \Big[3\ei(-Y^{-3}) } \def\eqfourdeetwo{ @@ -242,7 +242,7 @@ given our scaling ansatz, yielding \begin{align} &\begin{aligned} \eqthreedeeone\\ - &\hspace{8em} + &\hspace{7em} \eqthreedeetwo \end{aligned} \\ @@ -264,13 +264,13 @@ At the level of truncation of \eqref{eq:im.scaling} at which we are working the Kramers--Kronig relation does not converge in \twodee. However, higher moments can still be extracted, e.g., the susceptibility, by taking \[ - \chi=\pd[2]Fh - =\frac2\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. + \chi=\pd MH=-\frac1{T_\c}\pd[2]Fh + =-\frac2{\pi T_\c}\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. \] With a scaling form defined by $\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, this yields \[ - \fX^\twodee(Y/B)=\frac{AB^2}{\pi Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big] + \fX^\twodee(Y/B)=\frac{AB^2}{\pi T_\c Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big] \label{eq:sus_scaling} \] Scaling forms for the free energy can then be extracted by direct integration @@ -278,9 +278,9 @@ and their constants of integration fixed by known zero field values, yielding \begin{align} \label{eq:mag_scaling} \fM^\twodee(Y/B) - &=\fM(0)+\frac{ABT_\c}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\ + &=\fM(0)+\frac{AB}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\ \fF^\twodee(Y/B) - &=\fF(0)+T_\c Y\bigg(\frac{\fM(0)}B+\frac{AT_\c}\pi e^{1/Y}\ei(-1/Y)\bigg) + &=-Y\bigg(\frac{\fM(0)}B-\frac{A}\pi e^{1/Y}\ei(-1/Y)\bigg) \end{align} with $F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})+t^{2-\alpha}\log|t|$ in two dimensions. @@ -325,8 +325,8 @@ $C_0^-=0.025\,536\,971\,9$ \cite{barouch.1973.susceptibility}, so that $B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$. If we assume incorrectly that \eqref{eq:sus_scaling} is the true asymptotic form of the susceptibility scaling function, then -$\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ and the constant -$A$ is fixed to $A=\pi\fX(0)/2B^2=2^{11/8}\pi^3(\arcsinh1)^{19/4}C_0^-$. The +$\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi T_\c$ and the constant +$A$ is fixed to $A=\pi T_\c\fX(0)/2B^2=2^{19/8}\pi^3(\arcsinh1)^{15/4}C_0^-$. The resulting scaling functions $\fX$ and $\fM$ are plotted as solid lines in Fig.~\ref{fig:scaling_fits}. As can be seen, there is very good agreement between our proposed functional forms and what is measured. However, there @@ -349,8 +349,10 @@ where $F_n'(Y)=f_n(Y)$ and The functions $f_n$ have been chosen to be pure integer power laws for small argument, but vanish appropriately at large argument. This is necessary because the susceptibility vanishes with $h|t|^{-\beta\delta}$ while bare -polynomial corrections would not. We fit these functions to our numeric data -for $N=0$ while requiring that $C_0^-/T_\c=\fX'(0)=c_0+2AB^2/\pi$. The +polynomial corrections would not. We fit these functions to known moments of +the free energy's scaling function +\cite{mangazeev.2008.variational,mangazeev.2010.scaling} our numeric data +for $N=0$. The resulting curves are also plotted as dashed lines in Fig.~\ref{fig:scaling_fits}. Our singular scaling function with one low-order correction appears to match data quite well. @@ -385,10 +387,10 @@ into the scaling function gives good convergence to the simulations in \twodee. Our results should allow improved high-precision functional forms for the free energy~\cite{caselle.2001.critical}, and should have implications for the scaling -of correlation functions~\cite{YJXXX,XXX}. Our methods might be generalized +of correlation functions~\cite{chen.2013.universal,wu.1976.spin}. Our methods might be generalized to predict similar singularities in systems where nucleation and metastability are proximate to continuous phase transitions, such as 2D superfluid -transitions~\cite{ALHN}, the melting of 2D crystals~\cite{XXX}, and +transitions~\cite{ambegaokar.1978.dissipation,ambegaokar.1980.dynamics}, the melting of 2D crystals~\cite{XXX}, and freezing transitions in glasses, spin glasses, and other disordered systems. diff --git a/figs/fig-susmag.gplot b/figs/fig-susmag.gplot index 9682953..a2f4584 100644 --- a/figs/fig-susmag.gplot +++ b/figs/fig-susmag.gplot @@ -7,6 +7,8 @@ Delta = 15. / 8 gamma = 7. / 4 beta = 1. / 8 +Ch = 0.838677624411 + t(T) = abs((T - Tc) / T) h(T, H) = H / T X(T, H) = h(T, H) * t(T)**(-Delta) @@ -14,14 +16,69 @@ X(T, H) = h(T, H) * t(T)**(-Delta) poly(A, l, m, X) = A * X**m / (1 + (l * X)**(m + 1)) polyint(A, l, m, X) = A * l**(-(m + 1)) * log(1 + (l * X)**(m + 1)) / (m + 1) +G(i) = i == 1 ? -1.3578383417066 : \ + i == 2 ? -0.048953289720 : \ + i == 3 ? 0.038863932 : \ + i == 4 ? -0.068362119 : \ + i == 5 ? 0.18388370 : \ + i == 6 ? -0.6591714 : \ + i == 7 ? 2.937665 : \ + i == 8 ? -15.61 : 0 +GC(i) = G(i) * (2 * asinh(1))**2 * (Ch * (2 * asinh(1))**(-Delta))**i + M0 = (2**2.5 * asinh(1))**0.125 B = Tc**2 * M0 / (16 * pi) C0 = 0.0255369719 A = pi / 2 * C0 / (B**2 * Tc) -c0 = -0.012384 -lamb = 1.76962 -A2 = pi / 2 * (C0 / Tc - c0) / B**2 +#c0 = -0.012384 +#lamb = 1.76962 +#A2 = pi / 2 * (C0 / Tc - c0) / B**2 + +#c0 = 0 +#lamb = 1 +#B2 = 0.521944 +#A2 = pi / 2 * C0 / (B2**2 * Tc) + +n = 1 +c(i) = i == 1 ? 0.0037735 : 0 +lamb = 10.487 +B2 = B +A2 = 0.749317 + +#n = 2 +#c(i) = i == 1 ? -0.177238 : \ +# i == 2 ? -0.0545988 : 0 +#lamb = 3.14865 +#B2 = B +#A2 = 18.8816 + +#n = 3 +#c(i) = i == 1 ? 0.00280714 : \ +# i == 2 ? -0.00243938 : \ +# i == 3 ? -0.0140978 : 0 +#lamb = 12.1954 +#B2 = B +#A2 = 0.846118 + +n2 = 4 +c2(i) = i == 1 ? 0.00245324 : \ + i == 2 ? -0.00468448 : \ + i == 3 ? 0.0092602 : \ + i == 4 ? -0.536727 : 0 +lamb2 = 12.6068 +B3 = B +A3 = 0.88157 + +#n2 = 5 +#c2(i) = i == 1 ? 0.00281829 : \ +# i == 2 ? 0.000632215 : \ +# i == 3 ? -0.0911689 : \ +# i == 4 ? 1.93584 : \ +# i == 5 ? -24.7397 : 0 +#lamb2 = 13.2489 +#B3 = B +#A3 = 0.845002 susfunc = "figs/fig-sus_scaling-func.dat" magfunc = "figs/fig-mag_scaling-func.dat" @@ -42,7 +99,7 @@ set mxtics 5 set mytics 5 set bmargin 0.2 -plot num using (X($2, $3)):(10**3 * $10 * t($2)**gamma):(10**3 * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B):(10**(3+$2) * A2 * B**2 + 10**3 * poly(c0, lamb, 0, 10**$1)) with lines dt 2 lw 2 lc black +plot num using (X($2, $3)):(10**3 * $10 * t($2)**gamma):(10**3 * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B2):(10**(3+$2) * A2 * B2**2 + 10**3 * (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines dt 2 lw 2 lc black, susfunc using (10**$1 / B2):(10**(3+$2) * A3 * B3**2 + 10**3 * (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines dt 3 lw 2 lc black, susfunc using (10**$1 / B2):(-10**3 * (sum[i=1:7] GC(i + 1) * i * (i + 1) * (10**$1 / B2)**(i-1) / Tc)) with lines dt 5 lw 2 lc black set bmargin -1 set tmargin 0.2 @@ -53,7 +110,7 @@ set ylabel offset 1,0 '$M|t|^{-\beta}$' set xlabel '$h|t|^{-\beta\delta}$' set xtics format '%g' -plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A * B) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A2 * B + polyint(Tc * c0, lamb, 0, 10**$1) / B) smooth csplines with lines dt 2 lw 2 lc black +plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A * B) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B2):(M0 + 10**($2) * Tc * A2 * B2 + (sum[i=1:n] polyint(Tc * c(i), lamb, i-1, 10**$1)) / B2) smooth csplines with lines dt 2 lw 2 lc black, magfunc using (10**$1 / B3):(M0 + 10**($2) * Tc * A3 * B3 + (sum[i=1:n2] polyint(Tc * c2(i), lamb2, i-1, 10**$1)) / B3) smooth csplines with lines dt 3 lw 2 lc black, magfunc using (10**$1 / B):(-sum[i=1:8] GC(i) * i * (10**$1 / B)**(i-1)) with lines dt 5 lw 2 lc black set logscale xy set tmargin -1 @@ -70,7 +127,7 @@ set xtics add ('$\footnotesize10^{-2}$' 10**(-2), "" 0.1, '$\footnotesize10^0$' set mytics 5 set ytics format '\footnotesize$10^{%T}$' 0.00001,10,0.01 -plot num using (X($2, $3)):($10 * t($2)**gamma):($11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**$2 * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B):(10**$2 * A2 * B**2 + poly(c0, lamb, 0, 10**$1)) with lines dt 2 lw 2 lc black +plot num using (X($2, $3)):($10 * t($2)**gamma):($11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**$2 * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B2):(10**$2 * A2 * B2**2 + (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines dt 2 lw 2 lc black, susfunc using (10**$1 / B3):(10**$2 * A3 * B3**2 + (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines dt 3 lw 2 lc black, susfunc using (10**$1 / B):(-sum[i=2:8] GC(i) * i * (i-1) * (10**$1 / B)**(i-2) / Tc) with lines dt 5 lw 2 lc black unset logscale xy set logscale x @@ -81,5 +138,5 @@ set ylabel offset 4,0 '\footnotesize$M|t|^{-\beta}$' set ytics format '\footnotesize {%g}' 1.2,0.2,1.8 set mytics 5 -plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + Tc * A * B * 10**$2) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A2 * B + polyint(Tc * c0, lamb, 0, 10**$1) / B) with lines dt 2 lw 2 lc black +plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + Tc * A * B * 10**$2) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B2):(M0 + 10**($2) * Tc * A2 * B2 + (sum[i=1:n] polyint(Tc * c(i), lamb, i-1, 10**$1)) / B2) with lines dt 2 lw 2 lc black, magfunc using (10**$1 / B3):(M0 + 10**($2) * Tc * A3 * B3 + (sum[i=1:n2] polyint(Tc * c2(i), lamb2, i-1, 10**$1)) / B2) with lines dt 3 lw 2 lc black, magfunc using (10**$1 / B):(-sum[i=1:8] GC(i) * i * (10 **$1 / B)**(i-1)) with lines dt 5 lw 2 lc black -- cgit v1.2.3-70-g09d2 From 0c6de182fd0b2b933741bba6a79253a99c350488 Mon Sep 17 00:00:00 2001 From: "James P. Sethna" Date: Thu, 3 Aug 2017 12:30:20 -0400 Subject: Added ref and changed .gitignore so I can see the current version --- .gitignore | 1 - refs/PhysRevB.40.9006.pdf | Bin 0 -> 670094 bytes 2 files changed, 1 deletion(-) create mode 100644 refs/PhysRevB.40.9006.pdf diff --git a/.gitignore b/.gitignore index be37e5a..137e59e 100644 --- a/.gitignore +++ b/.gitignore @@ -1,4 +1,3 @@ -*.pdf *.aux *.log *.bbl diff --git a/refs/PhysRevB.40.9006.pdf b/refs/PhysRevB.40.9006.pdf new file mode 100644 index 0000000..d455145 Binary files /dev/null and b/refs/PhysRevB.40.9006.pdf differ -- cgit v1.2.3-70-g09d2 From 8c9c5ec6c859275137cf7826204967ba26576ef3 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 4 Aug 2017 09:09:36 -0400 Subject: some changes --- figs/fig-susmag.gplot | 46 +++++++++++++++++++++++----------------------- 1 file changed, 23 insertions(+), 23 deletions(-) diff --git a/figs/fig-susmag.gplot b/figs/fig-susmag.gplot index a2f4584..db33997 100644 --- a/figs/fig-susmag.gplot +++ b/figs/fig-susmag.gplot @@ -46,12 +46,12 @@ lamb = 10.487 B2 = B A2 = 0.749317 -#n = 2 -#c(i) = i == 1 ? -0.177238 : \ -# i == 2 ? -0.0545988 : 0 -#lamb = 3.14865 -#B2 = B -#A2 = 18.8816 +#n2 = 2 +#c2(i) = i == 1 ? -0.177238 : \ +# i == 2 ? -0.0545988 : 0 +#lamb2 = 3.14865 +#B3 = B +#A3 = 18.8816 #n = 3 #c(i) = i == 1 ? 0.00280714 : \ @@ -61,24 +61,24 @@ A2 = 0.749317 #B2 = B #A2 = 0.846118 -n2 = 4 -c2(i) = i == 1 ? 0.00245324 : \ - i == 2 ? -0.00468448 : \ - i == 3 ? 0.0092602 : \ - i == 4 ? -0.536727 : 0 -lamb2 = 12.6068 -B3 = B -A3 = 0.88157 - -#n2 = 5 -#c2(i) = i == 1 ? 0.00281829 : \ -# i == 2 ? 0.000632215 : \ -# i == 3 ? -0.0911689 : \ -# i == 4 ? 1.93584 : \ -# i == 5 ? -24.7397 : 0 -#lamb2 = 13.2489 +#n2 = 4 +#c2(i) = i == 1 ? 0.00245324 : \ +# i == 2 ? -0.00468448 : \ +# i == 3 ? 0.0092602 : \ +# i == 4 ? -0.536727 : 0 +#lamb2 = 12.6068 #B3 = B -#A3 = 0.845002 +#A3 = 0.88157 + +n2 = 5 +c2(i) = i == 1 ? 0.00281829 : \ + i == 2 ? 0.000632215 : \ + i == 3 ? -0.0911689 : \ + i == 4 ? 1.93584 : \ + i == 5 ? -24.7397 : 0 +lamb2 = 13.2489 +B3 = B +A3 = 0.845002 susfunc = "figs/fig-sus_scaling-func.dat" magfunc = "figs/fig-mag_scaling-func.dat" -- cgit v1.2.3-70-g09d2 From fc8d1b9c4678dbc5370641f6f7a75e29e6a82b04 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Sat, 5 Aug 2017 16:04:49 -0400 Subject: changed paper and figures to incorporate new definition of scaling functions --- .gitignore | 1 + essential-ising.tex | 11 +++++------ figs/fig-susmag.gplot | 34 +++++++++++++++++----------------- 3 files changed, 23 insertions(+), 23 deletions(-) diff --git a/.gitignore b/.gitignore index 137e59e..e413095 100644 --- a/.gitignore +++ b/.gitignore @@ -4,6 +4,7 @@ *.blg *Notes.bib *.dvi +essential-ising.pdf figs/*.tex fig-*.tex diff --git a/essential-ising.tex b/essential-ising.tex index c98e938..b7f432c 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -267,10 +267,10 @@ moments can still be extracted, e.g., the susceptibility, by taking \chi=\pd MH=-\frac1{T_\c}\pd[2]Fh =-\frac2{\pi T_\c}\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. \] -With a scaling form defined by $\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, +With a scaling form defined by $T_\c\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, this yields \[ - \fX^\twodee(Y/B)=\frac{AB^2}{\pi T_\c Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big] + \fX^\twodee(Y/B)=\frac{AB^2}{\pi Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big] \label{eq:sus_scaling} \] Scaling forms for the free energy can then be extracted by direct integration @@ -320,13 +320,12 @@ single curve, is plotted in Fig.~\ref{fig:scaling_fits}. For the \twodee Ising model on a square lattice, exact results at zero temperature have $\fS(0)=4/T_\c$, $\fM(0)=(2^{5/2}\arcsinh1)^\beta$ -\cite{onsager.1944.crystal}, and $\fX(0)=C_0^-/T_\c$ with -$C_0^-=0.025\,536\,971\,9$ \cite{barouch.1973.susceptibility}, so that +\cite{onsager.1944.crystal}, and $\fX(0)=C_0^-=0.025\,536\,971\,9$ \cite{barouch.1973.susceptibility}, so that $B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$. If we assume incorrectly that \eqref{eq:sus_scaling} is the true asymptotic form of the susceptibility scaling function, then -$\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi T_\c$ and the constant -$A$ is fixed to $A=\pi T_\c\fX(0)/2B^2=2^{19/8}\pi^3(\arcsinh1)^{15/4}C_0^-$. The +$T_\c\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ and the constant +$A$ is fixed to $A=\pi\fX(0)/2B^2=2^{19/8}\pi^3(\arcsinh1)^{15/4}C_0^-$. The resulting scaling functions $\fX$ and $\fM$ are plotted as solid lines in Fig.~\ref{fig:scaling_fits}. As can be seen, there is very good agreement between our proposed functional forms and what is measured. However, there diff --git a/figs/fig-susmag.gplot b/figs/fig-susmag.gplot index db33997..a639f4d 100644 --- a/figs/fig-susmag.gplot +++ b/figs/fig-susmag.gplot @@ -29,7 +29,7 @@ GC(i) = G(i) * (2 * asinh(1))**2 * (Ch * (2 * asinh(1))**(-Delta))**i M0 = (2**2.5 * asinh(1))**0.125 B = Tc**2 * M0 / (16 * pi) C0 = 0.0255369719 -A = pi / 2 * C0 / (B**2 * Tc) +A = pi / 2 * C0 / (B**2) #c0 = -0.012384 #lamb = 1.76962 @@ -41,10 +41,10 @@ A = pi / 2 * C0 / (B**2 * Tc) #A2 = pi / 2 * C0 / (B2**2 * Tc) n = 1 -c(i) = i == 1 ? 0.0037735 : 0 +c(i) = i == 1 ? 0.00856277 : 0 lamb = 10.487 B2 = B -A2 = 0.749317 +A2 = 1.70034 #n2 = 2 #c2(i) = i == 1 ? -0.177238 : \ @@ -71,14 +71,14 @@ A2 = 0.749317 #A3 = 0.88157 n2 = 5 -c2(i) = i == 1 ? 0.00281829 : \ - i == 2 ? 0.000632215 : \ - i == 3 ? -0.0911689 : \ - i == 4 ? 1.93584 : \ - i == 5 ? -24.7397 : 0 +c2(i) = i == 1 ? 0.00639522 : \ + i == 2 ? 0.00143461 : \ + i == 3 ? -0.206879 : \ + i == 4 ? 4.39278 : \ + i == 5 ? -56.1389 : 0 lamb2 = 13.2489 B3 = B -A3 = 0.845002 +A3 = 1.91747 susfunc = "figs/fig-sus_scaling-func.dat" magfunc = "figs/fig-mag_scaling-func.dat" @@ -91,15 +91,15 @@ set key off set size 1,1 - 0.52 set origin 0,0.52 set xrange [-40:1200] -set yrange [-0.4:14.4] -set ylabel offset 1,0 '$\chi|t|^{\gamma}\times 10^3$' +set yrange [-1:33] +set ylabel offset 1,0 '$T_\c\chi|t|^{\gamma}\times 10^3$' set lmargin 6 set xtics format '' set mxtics 5 set mytics 5 set bmargin 0.2 -plot num using (X($2, $3)):(10**3 * $10 * t($2)**gamma):(10**3 * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B2):(10**(3+$2) * A2 * B2**2 + 10**3 * (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines dt 2 lw 2 lc black, susfunc using (10**$1 / B2):(10**(3+$2) * A3 * B3**2 + 10**3 * (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines dt 3 lw 2 lc black, susfunc using (10**$1 / B2):(-10**3 * (sum[i=1:7] GC(i + 1) * i * (i + 1) * (10**$1 / B2)**(i-1) / Tc)) with lines dt 5 lw 2 lc black +plot num using (X($2, $3)):(10**3 * Tc * $10 * t($2)**gamma):(10**3 * Tc * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B2):(10**(3+$2) * A2 * B2**2 + 10**3 * (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines dt 2 lw 2 lc black, susfunc using (10**$1 / B2):(10**(3+$2) * A3 * B3**2 + 10**3 * (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines dt 3 lw 2 lc black, susfunc using (10**$1 / B2):(-10**3 * (sum[i=1:7] GC(i + 1) * i * (i + 1) * (10**$1 / B2)**(i-1))) with lines dt 5 lw 2 lc black set bmargin -1 set tmargin 0.2 @@ -110,7 +110,7 @@ set ylabel offset 1,0 '$M|t|^{-\beta}$' set xlabel '$h|t|^{-\beta\delta}$' set xtics format '%g' -plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A * B) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B2):(M0 + 10**($2) * Tc * A2 * B2 + (sum[i=1:n] polyint(Tc * c(i), lamb, i-1, 10**$1)) / B2) smooth csplines with lines dt 2 lw 2 lc black, magfunc using (10**$1 / B3):(M0 + 10**($2) * Tc * A3 * B3 + (sum[i=1:n2] polyint(Tc * c2(i), lamb2, i-1, 10**$1)) / B3) smooth csplines with lines dt 3 lw 2 lc black, magfunc using (10**$1 / B):(-sum[i=1:8] GC(i) * i * (10**$1 / B)**(i-1)) with lines dt 5 lw 2 lc black +plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + 10**($2) * A * B) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B2):(M0 + 10**($2) * A2 * B2 + (sum[i=1:n] polyint(c(i), lamb, i-1, 10**$1)) / B2) smooth csplines with lines dt 2 lw 2 lc black, magfunc using (10**$1 / B3):(M0 + 10**($2) * A3 * B3 + (sum[i=1:n2] polyint(c2(i), lamb2, i-1, 10**$1)) / B3) smooth csplines with lines dt 3 lw 2 lc black, magfunc using (10**$1 / B):(-sum[i=1:8] GC(i) * i * (10**$1 / B)**(i-1)) with lines dt 5 lw 2 lc black set logscale xy set tmargin -1 @@ -118,8 +118,8 @@ set lmargin -1 set size 0.65,0.325 set origin 0.31,0.5 + 0.29 / 2 set xrange [0.0015:1900] -set yrange [0.00002:0.02] -set ylabel offset 2.5,0 '\footnotesize$\chi|t|^\gamma$' +set yrange [0.00002:0.08] +set ylabel offset 2.5,0 '\footnotesize$T_\c\chi|t|^\gamma$' set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\beta\delta}$' set mxtics 5 set xtics format '' -2,10,1000 @@ -127,7 +127,7 @@ set xtics add ('$\footnotesize10^{-2}$' 10**(-2), "" 0.1, '$\footnotesize10^0$' set mytics 5 set ytics format '\footnotesize$10^{%T}$' 0.00001,10,0.01 -plot num using (X($2, $3)):($10 * t($2)**gamma):($11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**$2 * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B2):(10**$2 * A2 * B2**2 + (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines dt 2 lw 2 lc black, susfunc using (10**$1 / B3):(10**$2 * A3 * B3**2 + (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines dt 3 lw 2 lc black, susfunc using (10**$1 / B):(-sum[i=2:8] GC(i) * i * (i-1) * (10**$1 / B)**(i-2) / Tc) with lines dt 5 lw 2 lc black +plot num using (X($2, $3)):(Tc * $10 * t($2)**gamma):(Tc * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**$2 * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B2):(10**$2 * A2 * B2**2 + (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines dt 2 lw 2 lc black, susfunc using (10**$1 / B3):(10**$2 * A3 * B3**2 + (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines dt 3 lw 2 lc black, susfunc using (10**$1 / B):(-sum[i=2:8] GC(i) * i * (i-1) * (10**$1 / B)**(i-2)) with lines dt 5 lw 2 lc black unset logscale xy set logscale x @@ -138,5 +138,5 @@ set ylabel offset 4,0 '\footnotesize$M|t|^{-\beta}$' set ytics format '\footnotesize {%g}' 1.2,0.2,1.8 set mytics 5 -plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + Tc * A * B * 10**$2) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B2):(M0 + 10**($2) * Tc * A2 * B2 + (sum[i=1:n] polyint(Tc * c(i), lamb, i-1, 10**$1)) / B2) with lines dt 2 lw 2 lc black, magfunc using (10**$1 / B3):(M0 + 10**($2) * Tc * A3 * B3 + (sum[i=1:n2] polyint(Tc * c2(i), lamb2, i-1, 10**$1)) / B2) with lines dt 3 lw 2 lc black, magfunc using (10**$1 / B):(-sum[i=1:8] GC(i) * i * (10 **$1 / B)**(i-1)) with lines dt 5 lw 2 lc black +plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + A * B * 10**$2) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B2):(M0 + 10**($2) * A2 * B2 + (sum[i=1:n] polyint(c(i), lamb, i-1, 10**$1)) / B2) with lines dt 2 lw 2 lc black, magfunc using (10**$1 / B3):(M0 + 10**($2) * A3 * B3 + (sum[i=1:n2] polyint(c2(i), lamb2, i-1, 10**$1)) / B2) with lines dt 3 lw 2 lc black, magfunc using (10**$1 / B):(-sum[i=1:8] GC(i) * i * (10 **$1 / B)**(i-1)) with lines dt 5 lw 2 lc black -- cgit v1.2.3-70-g09d2 From de6530d03dcff6302510bdc19fdb04fdc0572ae1 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Sat, 5 Aug 2017 21:15:12 -0400 Subject: added new plot, made many changes --- essential-ising.tex | 8 ++++ figs/fig-series-data.dat | 100 +++++++++++++++++++++++++++++++++++++++++++++++ figs/fig-series.gplot | 13 ++++++ figs/fig-susmag.gplot | 41 ++----------------- makefile | 4 +- 5 files changed, 127 insertions(+), 39 deletions(-) create mode 100644 figs/fig-series-data.dat create mode 100644 figs/fig-series.gplot diff --git a/essential-ising.tex b/essential-ising.tex index b7f432c..9397aa0 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -374,6 +374,14 @@ correction appears to match data quite well. \label{fig:scaling_fits} \end{figure} +\begin{figure} + \input{fig-series} + \caption{ + Something + } + \label{fig:series} +\end{figure} + Abrupt phase transitions, such as the jump in magnetization in the Ising model below $T_\c$, are known to imply essential singularities in the free energy that are usually thought to be unobservable in practice. We have diff --git a/figs/fig-series-data.dat b/figs/fig-series-data.dat new file mode 100644 index 0000000..b59d9e5 --- /dev/null +++ b/figs/fig-series-data.dat @@ -0,0 +1,100 @@ +1. 1.222409953716735 1.222409953716735 1.222409953716735 1.2224099537167044 +2. 0.012768485949999997 0.012768487260939602 0.012768487260939605 0.012768487260939603 +3. 0.0015989185962086278 0.002936926367989711 0.0029369263679897106 0.002936926367989711 +4. 0.00040044539146033526 0.0014967545682344 0.0014967545682344 0.0014967545682344006 +5. 0.00015043590579476038 0.001069611169121874 0.001166452519998439 0.0011664525199984396 +6. 0.00007535263562324922 0.0008989696219143944 0.0012114652004510606 0.0012114652004510617 +7. 0.00004717972469216129 0.0008276269886291274 0.0015642408094569038 0.0015642408094569044 +8. 0.00003544815233522661 0.0008078196901937446 0.0024082024927311986 0.002408202492731202 +9. 0.00003107267152455227 0.0008216915431728964 0.004056269379287904 0. +10. 0.00003112830920066225 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+81. 5.957255694957615e43 3.9597362041031383e43 4.465381522762211e43 0. +82. 5.967922555790579e44 3.966826374004722e44 4.473377084092526e44 0. +83. 6.053341122798547e45 4.0236033548165993e45 4.5374042990299425e45 0. +84. 6.215784531727004e46 4.1315780801580665e46 4.659167042458534e46 0. +85. 6.460423592021501e47 4.29418754223065e47 4.8425411992048355e47 0. +86. 6.795590978710545e48 4.5169703978718155e48 5.093772694408335e48 0. +87. 7.233243867785832e49 4.807874478280617e49 5.421824271229932e49 0. +88. 7.789660130188514e50 5.177719543123572e50 5.838897337138516e50 0. +89. 8.48642373535593e51 5.6408522697275565e51 6.361170593074344e51 0. +90. 9.351781162961525e52 6.216047848201684e52 7.009816759309207e52 0. +91. 1.0422485337917562e54 6.927735629044529e53 7.812384734230928e53 0. +92. 1.174629092414621e55 7.807657723276689e54 8.804669992269844e54 0. +93. 1.3385330408368046e56 8.897113056060406e55 1.0033245195801371e56 0. +94. 1.5420692388222914e57 1.0250000515114435e57 1.1558891943626807e57 0. +95. 1.7958653585021562e58 1.1936961315551265e58 1.3461272102206315e58 0. +96. 2.1139201448674616e59 1.4051043901472477e59 1.584531609659763e59 0. +97. 2.5147749949180078e60 1.671549133098101e60 1.885000471897858e60 0. +98. 3.0231334393007486e61 2.0094505830203726e61 2.266050828089256e61 0. +99. 3.6721127039111263e62 2.440821472801901e62 2.7525063648727253e62 0. +100. 4.50639263448796e63 2.9953601084790524e63 3.377857764505008e63 0. \ No newline at end of file diff --git a/figs/fig-series.gplot b/figs/fig-series.gplot new file mode 100644 index 0000000..bde0cae --- /dev/null +++ b/figs/fig-series.gplot @@ -0,0 +1,13 @@ + +set terminal pslatex rotate size 3.417,2.111 + +set logscale y + +data = "figs/fig-series-data.dat" + +set xrange [0:20] +set yrange [0.000001:10] +set key off + +plot data using 1:2 with points, data using 1:3 with points, data using 1:4 with points, data using 1:5 with points + diff --git a/figs/fig-susmag.gplot b/figs/fig-susmag.gplot index a639f4d..80679b6 100644 --- a/figs/fig-susmag.gplot +++ b/figs/fig-susmag.gplot @@ -31,44 +31,11 @@ B = Tc**2 * M0 / (16 * pi) C0 = 0.0255369719 A = pi / 2 * C0 / (B**2) -#c0 = -0.012384 -#lamb = 1.76962 -#A2 = pi / 2 * (C0 / Tc - c0) / B**2 - -#c0 = 0 -#lamb = 1 -#B2 = 0.521944 -#A2 = pi / 2 * C0 / (B2**2 * Tc) - n = 1 c(i) = i == 1 ? 0.00856277 : 0 lamb = 10.487 -B2 = B A2 = 1.70034 - -#n2 = 2 -#c2(i) = i == 1 ? -0.177238 : \ -# i == 2 ? -0.0545988 : 0 -#lamb2 = 3.14865 -#B3 = B -#A3 = 18.8816 - -#n = 3 -#c(i) = i == 1 ? 0.00280714 : \ -# i == 2 ? -0.00243938 : \ -# i == 3 ? -0.0140978 : 0 -#lamb = 12.1954 -#B2 = B -#A2 = 0.846118 - -#n2 = 4 -#c2(i) = i == 1 ? 0.00245324 : \ -# i == 2 ? -0.00468448 : \ -# i == 3 ? 0.0092602 : \ -# i == 4 ? -0.536727 : 0 -#lamb2 = 12.6068 -#B3 = B -#A3 = 0.88157 +B2 = B n2 = 5 c2(i) = i == 1 ? 0.00639522 : \ @@ -77,8 +44,8 @@ c2(i) = i == 1 ? 0.00639522 : \ i == 4 ? 4.39278 : \ i == 5 ? -56.1389 : 0 lamb2 = 13.2489 -B3 = B A3 = 1.91747 +B3 = B susfunc = "figs/fig-sus_scaling-func.dat" magfunc = "figs/fig-mag_scaling-func.dat" @@ -99,7 +66,7 @@ set mxtics 5 set mytics 5 set bmargin 0.2 -plot num using (X($2, $3)):(10**3 * Tc * $10 * t($2)**gamma):(10**3 * Tc * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B2):(10**(3+$2) * A2 * B2**2 + 10**3 * (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines dt 2 lw 2 lc black, susfunc using (10**$1 / B2):(10**(3+$2) * A3 * B3**2 + 10**3 * (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines dt 3 lw 2 lc black, susfunc using (10**$1 / B2):(-10**3 * (sum[i=1:7] GC(i + 1) * i * (i + 1) * (10**$1 / B2)**(i-1))) with lines dt 5 lw 2 lc black +plot num using (X($2, $3)):(10**3 * Tc * $10 * t($2)**gamma):(10**3 * Tc * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B2):(10**(3+$2) * A2 * B2**2 + 10**3 * (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines dt 2 lw 2 lc black, susfunc using (10**$1 / B3):(10**(3+$2) * A3 * B3**2 + 10**3 * (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines dt 3 lw 2 lc black, susfunc using (10**$1 / B3):(-10**3 * (sum[i=1:7] GC(i + 1) * i * (i + 1) * (10**$1 / B3)**(i-1))) with lines dt 5 lw 2 lc black set bmargin -1 set tmargin 0.2 @@ -138,5 +105,5 @@ set ylabel offset 4,0 '\footnotesize$M|t|^{-\beta}$' set ytics format '\footnotesize {%g}' 1.2,0.2,1.8 set mytics 5 -plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + A * B * 10**$2) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B2):(M0 + 10**($2) * A2 * B2 + (sum[i=1:n] polyint(c(i), lamb, i-1, 10**$1)) / B2) with lines dt 2 lw 2 lc black, magfunc using (10**$1 / B3):(M0 + 10**($2) * A3 * B3 + (sum[i=1:n2] polyint(c2(i), lamb2, i-1, 10**$1)) / B2) with lines dt 3 lw 2 lc black, magfunc using (10**$1 / B):(-sum[i=1:8] GC(i) * i * (10 **$1 / B)**(i-1)) with lines dt 5 lw 2 lc black +plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + A * B * 10**$2) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B2):(M0 + 10**($2) * A2 * B2 + (sum[i=1:n] polyint(c(i), lamb, i-1, 10**$1)) / B2) with lines dt 2 lw 2 lc black, magfunc using (10**$1 / B3):(M0 + 10**($2) * A3 * B3 + (sum[i=1:n2] polyint(c2(i), lamb2, i-1, 10**$1)) / B3) with lines dt 3 lw 2 lc black, magfunc using (10**$1 / B):(-sum[i=1:8] GC(i) * i * (10 **$1 / B)**(i-1)) with lines dt 5 lw 2 lc black diff --git a/makefile b/makefile index 32f4b74..a04aaa6 100644 --- a/makefile +++ b/makefile @@ -1,8 +1,8 @@ DOC=essential-ising -FIGS=fig-susmag +FIGS=fig-susmag fig-series DATA=data_square-4096 -FUNCS=fig-mag_scaling-func fig-sus_scaling-func +FUNCS=fig-mag_scaling-func fig-sus_scaling-func fig-series-data all: ${DOC}.pdf -- cgit v1.2.3-70-g09d2 From e1f268865e9148f8d4fdeddeef8d591a7ce86f99 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 7 Aug 2017 14:01:30 -0400 Subject: many changes --- essential-ising.bib | 1 - essential-ising.tex | 20 ++++++++++++++------ figs/fig-series.gplot | 19 ++++++++++++++++--- figs/fig-susmag.gplot | 36 +++++++++++++++++++++++++++++++----- 4 files changed, 61 insertions(+), 15 deletions(-) diff --git a/essential-ising.bib b/essential-ising.bib index 16a0883..f490dd9 100644 --- a/essential-ising.bib +++ b/essential-ising.bib @@ -115,7 +115,6 @@ year={2013} } - @article{dimitrovic.1991.finite, title={Finite-size effects, goldstone bosons and critical exponents in the d= 3 Heisenberg model}, author={Dimitrovi{\'c}, I and Hasenfratz, P and Nager, J and Niedermayer, Ferenc}, diff --git a/essential-ising.tex b/essential-ising.tex index 9397aa0..ba8e452 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -281,6 +281,7 @@ and their constants of integration fixed by known zero field values, yielding &=\fM(0)+\frac{AB}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\ \fF^\twodee(Y/B) &=-Y\bigg(\frac{\fM(0)}B-\frac{A}\pi e^{1/Y}\ei(-1/Y)\bigg) + \label{eq:2d_free_scaling} \end{align} with $F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})+t^{2-\alpha}\log|t|$ in two dimensions. @@ -364,12 +365,13 @@ correction appears to match data quite well. $h|t|^{-\beta\delta}$. Points with error bars show data with sampling error taken from simulations of a $4096\times4096$ square-lattice Ising model with periodic boundary conditions and $T_\c-T=0.01,0.02,\ldots,0.1$ - and $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Color denotes the value of - $T$, with red corresponding to $0.01$ and violet to $0.1$. The solid lines + and $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. The solid blue lines show our analytic results \eqref{eq:sus_scaling} and - \eqref{eq:mag_scaling}, while the dashed lines show fits of - \eqref{eq:sus_scaling_poly} and \eqref{eq:mag_scaling_poly} to the data - for $N=0$, with $c_0=-0.0124$ and $\lambda=1.77$. + \eqref{eq:mag_scaling}, the dashed yellow lines show + \eqref{eq:sus_scaling_poly} and \eqref{eq:mag_scaling_poly} for $N=0$, the + dotted green lines show the same for $N=4$, and the red line show the + polynomial resulting from truncating the series after the eight known + terms. } \label{fig:scaling_fits} \end{figure} @@ -377,7 +379,13 @@ correction appears to match data quite well. \begin{figure} \input{fig-series} \caption{ - Something + The series coefficients defined by $\tilde\fF(X)=\sum_nf_nX^n$. The blue + pluses correspond to the scaling form \eqref{eq:2d_free_scaling}, the + yellow saltires correspond to that form with the first four coefficients + fixed to known values (\eqref{eq:sus_scaling_poly} with $N=0$), the green + stars correspond to that form with the first eight coefficients fixed to + known values (\eqref{eq:sus_scaling_poly} with $N=4$), and the red squares + correspond to the first eight coefficients. } \label{fig:series} \end{figure} diff --git a/figs/fig-series.gplot b/figs/fig-series.gplot index bde0cae..b9cfc57 100644 --- a/figs/fig-series.gplot +++ b/figs/fig-series.gplot @@ -1,13 +1,26 @@ set terminal pslatex rotate size 3.417,2.111 +cc1 = "#5e81b5" +cc2 = "#e19c24" +cc3 = "#8fb032" +cc4 = "#eb6235" + set logscale y data = "figs/fig-series-data.dat" -set xrange [0:20] -set yrange [0.000001:10] +set xrange [0:30] +set yrange [0.000004:600] set key off +set xlabel '$n$' +set ylabel offset 2 '$|f_n|$' + +set ytics format '\footnotesize$10^{%T}$' 0.00001,10,1000 -plot data using 1:2 with points, data using 1:3 with points, data using 1:4 with points, data using 1:5 with points +plot \ + data using 1:2 with points lc rgb cc1, \ + data using 1:3 with points lc rgb cc2, \ + data using 1:4 with points lc rgb cc3, \ + data using 1:5 with points lc rgb cc4 diff --git a/figs/fig-susmag.gplot b/figs/fig-susmag.gplot index 80679b6..c56c898 100644 --- a/figs/fig-susmag.gplot +++ b/figs/fig-susmag.gplot @@ -2,6 +2,11 @@ set terminal pslatex rotate size 3.417,4.222 set multiplot +cc1 = "#5e81b5" +cc2 = "#e19c24" +cc3 = "#8fb032" +cc4 = "#eb6235" + Tc = 2 / log(1 + sqrt(2)) Delta = 15. / 8 gamma = 7. / 4 @@ -52,7 +57,7 @@ magfunc = "figs/fig-mag_scaling-func.dat" num = "data/data_square-4096.dat" -set samples 10000 +set samples 20000 set key off set size 1,1 - 0.52 @@ -66,7 +71,12 @@ set mxtics 5 set mytics 5 set bmargin 0.2 -plot num using (X($2, $3)):(10**3 * Tc * $10 * t($2)**gamma):(10**3 * Tc * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B2):(10**(3+$2) * A2 * B2**2 + 10**3 * (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines dt 2 lw 2 lc black, susfunc using (10**$1 / B3):(10**(3+$2) * A3 * B3**2 + 10**3 * (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines dt 3 lw 2 lc black, susfunc using (10**$1 / B3):(-10**3 * (sum[i=1:7] GC(i + 1) * i * (i + 1) * (10**$1 / B3)**(i-1))) with lines dt 5 lw 2 lc black +plot \ + num using (X($2, $3)):(10**3 * Tc * $10 * t($2)**gamma):(10**3 * Tc * $11 * t($2)**gamma) with yerrorbars pt 0 lc black, \ + susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb cc1 dt 1, \ + susfunc using (10**$1 / B2):(10**(3+$2) * A2 * B2**2 + 10**3 * (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines lw 2 lc rgb cc2 dt 2, \ + susfunc using (10**$1 / B3):(10**(3+$2) * A3 * B3**2 + 10**3 * (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines lw 2 lc rgb cc3 dt 3, \ + -10**3 * (sum[i=1:7] GC(i + 1) * i * (i + 1) * (x)**(i-1)) with lines lw 2 lc rgb cc4 dt 4 set bmargin -1 set tmargin 0.2 @@ -77,7 +87,12 @@ set ylabel offset 1,0 '$M|t|^{-\beta}$' set xlabel '$h|t|^{-\beta\delta}$' set xtics format '%g' -plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + 10**($2) * A * B) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B2):(M0 + 10**($2) * A2 * B2 + (sum[i=1:n] polyint(c(i), lamb, i-1, 10**$1)) / B2) smooth csplines with lines dt 2 lw 2 lc black, magfunc using (10**$1 / B3):(M0 + 10**($2) * A3 * B3 + (sum[i=1:n2] polyint(c2(i), lamb2, i-1, 10**$1)) / B3) smooth csplines with lines dt 3 lw 2 lc black, magfunc using (10**$1 / B):(-sum[i=1:8] GC(i) * i * (10**$1 / B)**(i-1)) with lines dt 5 lw 2 lc black +plot \ + num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)) with yerrorbars pt 0 lc black, \ + magfunc using (10**$1 / B):(M0 + 10**($2) * A * B) with linespoints pt 0 lw 2 lc rgb cc1 dt 1, \ + magfunc using (10**$1 / B2):(M0 + 10**($2) * A2 * B2 + (sum[i=1:n] polyint(c(i), lamb, i-1, 10**$1)) / B2) smooth csplines with lines lw 2 lc rgb cc2 dt 2, \ + magfunc using (10**$1 / B3):(M0 + 10**($2) * A3 * B3 + (sum[i=1:n2] polyint(c2(i), lamb2, i-1, 10**$1)) / B3) smooth csplines with lines lw 2 lc rgb cc3 dt 3, \ + -sum[i=1:8] GC(i) * i * x**(i-1) with lines lw 2 lc rgb cc4 dt 4 set logscale xy set tmargin -1 @@ -94,7 +109,13 @@ set xtics add ('$\footnotesize10^{-2}$' 10**(-2), "" 0.1, '$\footnotesize10^0$' set mytics 5 set ytics format '\footnotesize$10^{%T}$' 0.00001,10,0.01 -plot num using (X($2, $3)):(Tc * $10 * t($2)**gamma):(Tc * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**$2 * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B2):(10**$2 * A2 * B2**2 + (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines dt 2 lw 2 lc black, susfunc using (10**$1 / B3):(10**$2 * A3 * B3**2 + (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines dt 3 lw 2 lc black, susfunc using (10**$1 / B):(-sum[i=2:8] GC(i) * i * (i-1) * (10**$1 / B)**(i-2)) with lines dt 5 lw 2 lc black +plot \ + num using (X($2, $3)):(Tc * $10 * t($2)**gamma):(Tc * $11 * t($2)**gamma) with yerrorbars pt 0 lc black, \ + susfunc using (10**$1 / B):(10**$2 * A * B**2) with linespoints pt 0 lw 2 lc rgb cc1 dt 1, \ + susfunc using (10**$1 / B2):(10**$2 * A2 * B2**2 + (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines lw 2 lc rgb cc2 dt 2, \ + susfunc using (10**$1 / B3):(10**$2 * A3 * B3**2 + (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines lw 2 lc rgb cc3 dt 3, \ + -sum[i=2:8] GC(i) * i * (i-1) * x**(i-2) with lines dt 4 lw 2 lc rgb cc4, \ + 0.0705991 * x**(-14./15) unset logscale xy set logscale x @@ -105,5 +126,10 @@ set ylabel offset 4,0 '\footnotesize$M|t|^{-\beta}$' set ytics format '\footnotesize {%g}' 1.2,0.2,1.8 set mytics 5 -plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + A * B * 10**$2) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B2):(M0 + 10**($2) * A2 * B2 + (sum[i=1:n] polyint(c(i), lamb, i-1, 10**$1)) / B2) with lines dt 2 lw 2 lc black, magfunc using (10**$1 / B3):(M0 + 10**($2) * A3 * B3 + (sum[i=1:n2] polyint(c2(i), lamb2, i-1, 10**$1)) / B3) with lines dt 3 lw 2 lc black, magfunc using (10**$1 / B):(-sum[i=1:8] GC(i) * i * (10 **$1 / B)**(i-1)) with lines dt 5 lw 2 lc black +plot \ + num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)) with yerrorbars pt 0 lc black, \ + magfunc using (10**$1 / B):(M0 + A * B * 10**$2) with linespoints pt 0 lw 2 lc rgb cc1 dt 1, \ + magfunc using (10**$1 / B2):(M0 + 10**($2) * A2 * B2 + (sum[i=1:n] polyint(c(i), lamb, i-1, 10**$1)) / B2) with lines lw 2 lc rgb cc2 dt 2, \ + magfunc using (10**$1 / B3):(M0 + 10**($2) * A3 * B3 + (sum[i=1:n2] polyint(c2(i), lamb2, i-1, 10**$1)) / B3) with lines lw 2 lc rgb cc3 dt 3, \ + -sum[i=1:8] GC(i) * i * x**(i-1) with lines lw 2 lc rgb cc4 dt 4 -- cgit v1.2.3-70-g09d2 From c1fd415bd5dcee285c95e36e4def5fa412a87bb1 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 22 Aug 2017 22:28:01 -0400 Subject: new citations, some small fixes --- essential-ising.bib | 73 +++++++++++++++++++++++++++++++++++++++++++++++++++++ essential-ising.tex | 19 ++++++++------ 2 files changed, 84 insertions(+), 8 deletions(-) diff --git a/essential-ising.bib b/essential-ising.bib index f490dd9..1de2478 100644 --- a/essential-ising.bib +++ b/essential-ising.bib @@ -28,6 +28,13 @@ publisher={APS} } +@article{assis.2017.analyticity, + title={Analyticity of the Ising susceptibility: An interpretation}, + author={Assis, M and Jacobsen, JL and Jensen, I and Maillard, JM and McCoy, BM}, + journal={arXiv preprint arXiv:1705.02541}, + year={2017} +} + @article{barouch.1973.susceptibility, title={Zero-field susceptibility of the two-dimensional Ising model near ${T}_c$}, author={Barouch, Eytan and McCoy, Barry M and Wu, Tai Tsun}, @@ -108,6 +115,17 @@ publisher={IOP Publishing} } +@article{chan.2011.ising, + title={The Ising susceptibility scaling function}, + author={Chan, Y and Guttmann, Anthony J and Nickel, BG and Perk, JHH}, + journal={Journal of Statistical Physics}, + volume={145}, + number={3}, + pages={549--590}, + year={2011}, + publisher={Springer} +} + @article{chen.2013.universal, title={Universal scaling function for the two-dimensional Ising model in an external field: A pragmatic approach}, author={Chen, Yan-Jiun and Paquette, Natalie M and Machta, Benjamin B and Sethna, James P}, @@ -115,6 +133,17 @@ year={2013} } +@article{dahm.1989.dynamics, + title={Dynamics of dislocation-mediated melting in a two-dimensional lattice in the presence of an oscillatory applied strain}, + author={Dahm, AJ and Stan, MA and Petschek, RG}, + journal={Physical Review B}, + volume={40}, + number={13}, + pages={9006}, + year={1989}, + publisher={APS} +} + @article{dimitrovic.1991.finite, title={Finite-size effects, goldstone bosons and critical exponents in the d= 3 Heisenberg model}, author={Dimitrovi{\'c}, I and Hasenfratz, P and Nager, J and Niedermayer, Ferenc}, @@ -221,6 +250,17 @@ publisher={APS} } +@article{guttmann.1996.solvability, + title={Solvability of some statistical mechanical systems}, + author={Guttmann, AJ and Enting, IG}, + journal={Physical review letters}, + volume={76}, + number={3}, + pages={344}, + year={1996}, + publisher={APS} +} + @article{harris.1984.metastability, title={Metastability in the (1+ 1) D Ising model: a primitive droplet model calculation}, author={Harris, CK}, @@ -363,6 +403,28 @@ publisher={AIP} } +@article{nickel.1999.singularity, + title={On the singularity structure of the 2D Ising model susceptibility}, + author={Nickel, Bernie}, + journal={Journal of Physics A: Mathematical and General}, + volume={32}, + number={21}, + pages={3889}, + year={1999}, + publisher={IOP Publishing} +} + +@article{nickel.2000.addendum, + title={Addendum toOn the singularity structure of the 2D Ising model susceptibility'}, + author={Nickel, Bernie}, + journal={Journal of Physics A: Mathematical and General}, + volume={33}, + number={8}, + pages={1693}, + year={2000}, + publisher={IOP Publishing} +} + @article{onsager.1944.crystal, title={Crystal statistics. I. A two-dimensional model with an order-disorder transition}, author={Onsager, Lars}, @@ -374,6 +436,17 @@ publisher={APS} } +@article{orrick.2001.susceptibility, + title={The susceptibility of the square lattice Ising model: new developments}, + author={Orrick, WP and Nickel, B and Guttmann, AJ and Perk, JHH}, + journal={Journal of Statistical Physics}, + volume={102}, + number={3}, + pages={795--841}, + year={2001}, + publisher={Springer} +} + @article{parisi.1977.asymptotic, title={Asymptotic estimates in perturbation theory}, author={Parisi, Giorgio}, diff --git a/essential-ising.tex b/essential-ising.tex index ba8e452..d641a57 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -119,8 +119,10 @@ methods of critical droplet theory for the decay of an Ising system in a metastable state, i.e., an equilibrium Ising state for $T0$ subjected to a small negative external field $H<0$. The existence of an essential singularity has also been suggested by transfer matrix -\cite{mccraw.1978.metastability,enting.1980.investigation} and \textsc{rg} -methods \cite{klein.1976.essential}. It has long been known that the decay +\cite{mccraw.1978.metastability,enting.1980.investigation,mangazeev.2008.variational,mangazeev.2010.scaling} and \textsc{rg} +methods \cite{klein.1976.essential}, and a different kind of essential +singularity is known to exist in the zero-temperature susceptibility +\cite{orrick.2001.susceptibility,chan.2011.ising,guttmann.1996.solvability,nickel.1999.singularity,nickel.2000.addendum,assis.2017.analyticity}. It has long been known that the decay rate $\Gamma$ of metastable states in statistical mechanics is often related to the metastable free energy $F$ by $\Gamma\propto\im F$ \cite{langer.1969.metastable,penrose.1987.rigorous,gaveau.1989.analytic,privman.1982.analytic}. @@ -264,10 +266,10 @@ At the level of truncation of \eqref{eq:im.scaling} at which we are working the Kramers--Kronig relation does not converge in \twodee. However, higher moments can still be extracted, e.g., the susceptibility, by taking \[ - \chi=\pd MH=-\frac1{T_\c}\pd[2]Fh - =-\frac2{\pi T_\c}\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. + \chi=\pd MH=-\frac1{T}\pd[2]Fh + =-\frac2{\pi T}\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. \] -With a scaling form defined by $T_\c\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, +With a scaling form defined by $T\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, this yields \[ \fX^\twodee(Y/B)=\frac{AB^2}{\pi Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big] @@ -325,7 +327,7 @@ temperature have $\fS(0)=4/T_\c$, $\fM(0)=(2^{5/2}\arcsinh1)^\beta$ $B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$. If we assume incorrectly that \eqref{eq:sus_scaling} is the true asymptotic form of the susceptibility scaling function, then -$T_\c\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ and the constant +$T\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ and the constant $A$ is fixed to $A=\pi\fX(0)/2B^2=2^{19/8}\pi^3(\arcsinh1)^{15/4}C_0^-$. The resulting scaling functions $\fX$ and $\fM$ are plotted as solid lines in Fig.~\ref{fig:scaling_fits}. As can be seen, there is very good agreement @@ -405,7 +407,7 @@ energy~\cite{caselle.2001.critical}, and should have implications for the scalin of correlation functions~\cite{chen.2013.universal,wu.1976.spin}. Our methods might be generalized to predict similar singularities in systems where nucleation and metastability are proximate to continuous phase transitions, such as 2D superfluid -transitions~\cite{ambegaokar.1978.dissipation,ambegaokar.1980.dynamics}, the melting of 2D crystals~\cite{XXX}, and +transitions~\cite{ambegaokar.1978.dissipation,ambegaokar.1980.dynamics}, the melting of 2D crystals~\cite{dahm.1989.dynamics}, and freezing transitions in glasses, spin glasses, and other disordered systems. @@ -424,7 +426,8 @@ freezing transitions in glasses, spin glasses, and other disordered systems. \begin{acknowledgments} The authors would like to thank Tom Lubensky, Andrea Liu, and Randy Kamien - for helpful conversations. JPS thanks Jim Langer for past inspiration, + for helpful conversations. The authors would also like to think Jacques Perk + for pointing us to several insightful studies. JPS thanks Jim Langer for past inspiration, guidance, and encouragement. This work was supported by NSF grants DMR-1312160 and DMR-1719490. \end{acknowledgments} -- cgit v1.2.3-70-g09d2 From 80c08bc131afb141d0ed780bd7ebb939f4303dbd Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 26 Sep 2017 07:20:52 -0400 Subject: changed figures, added new matching analysis --- figs/fig-mag_scaling-func-mod.dat | 5498 ++++++++++++++++++++++++++++++++++++ figs/fig-series-data.dat | 107 +- figs/fig-series.gplot | 9 +- figs/fig-sus_scaling-func-mod.dat | 5499 +++++++++++++++++++++++++++++++++++++ figs/fig-susmag.gplot | 27 +- 5 files changed, 11020 insertions(+), 120 deletions(-) create mode 100644 figs/fig-mag_scaling-func-mod.dat create mode 100644 figs/fig-sus_scaling-func-mod.dat diff --git a/figs/fig-mag_scaling-func-mod.dat b/figs/fig-mag_scaling-func-mod.dat new file mode 100644 index 0000000..95d94f2 --- /dev/null +++ b/figs/fig-mag_scaling-func-mod.dat @@ -0,0 +1,5498 @@ +0.0029245327965242416 1.22241 +0.004878604313566581 1.2224597066045628 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3.0231334393007486e61 2.0094505830203726e61 2.266050828089256e61 0. -99. 3.6721127039111263e62 2.440821472801901e62 2.7525063648727253e62 0. -100. 4.50639263448796e63 2.9953601084790524e63 3.377857764505008e63 0. \ No newline at end of file +0 0.025536974521879204 0.025536974521879207 0.025536974521879207 +1 0.009593512562216971 0.01762155820793826 0.017621558207938266 +2 0.004805345190888446 0.01681987670730995 0.017961054818812806 +3 0.003008718424800099 0.018814565927230777 0.02332905039996879 +4 0.0022605793007909824 0.02318633276267007 0.03634395601353185 +5 0.0019815486405162243 0.0306005114760245 0.06569811399718999 +6 0.001985096734582422 0.04264499878327929 0.13485933959294732 \ No newline at end of file diff --git a/figs/fig-series.gplot b/figs/fig-series.gplot index b9cfc57..171af71 100644 --- a/figs/fig-series.gplot +++ b/figs/fig-series.gplot @@ -10,17 +10,16 @@ set logscale y data = "figs/fig-series-data.dat" -set xrange [0:30] -set yrange [0.000004:600] +set xrange [-0.5:6.5] +set yrange [0.0006:0.4] set key off set xlabel '$n$' set ylabel offset 2 '$|f_n|$' -set ytics format '\footnotesize$10^{%T}$' 0.00001,10,1000 +#set ytics format '\footnotesize$10^{%T}$' 0.001,10,1 plot \ data using 1:2 with points lc rgb cc1, \ data using 1:3 with points lc rgb cc2, \ - data using 1:4 with points lc rgb cc3, \ - data using 1:5 with points lc rgb cc4 + data using 1:4 with points lc rgb cc3 diff --git a/figs/fig-sus_scaling-func-mod.dat b/figs/fig-sus_scaling-func-mod.dat new file mode 100644 index 0000000..1845c84 --- /dev/null +++ b/figs/fig-sus_scaling-func-mod.dat @@ -0,0 +1,5499 @@ +0.0019484365898063418 0.02550279738908768 +0.0039006290032421414 0.025468497740770927 +0.00585657962389102 0.025434387321417607 +0.007816290839447943 0.025400325951561654 +0.009779765041686712 0.02536633675961013 +0.011747004626349592 0.025332417172490865 +0.013718011993166425 0.025298564980874367 +0.01569278954587821 0.0252647817513512 +0.017671339692470776 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1.6812086864888827e-7 +1.11906770958036e6 1.594901776076466e-7 +1.1861571640685692e6 1.510595015735598e-7 +1.2595740110830024e6 1.4283005386941177e-7 +1.3401484669727264e6 1.3480310448182895e-7 +1.428852495117011e6 1.2697998430884265e-7 +1.5268300178193366e6 1.193620898794402e-7 +1.6354349450412903e6 1.119508886197746e-7 +1.7562794268101787e6 1.0474792474595772e-7 +1.8912956049093255e6 9.775482588816365e-8 +2.0428153722401739e6 9.09733105698473e-8 +2.2136744199272124e6 8.440519668869089e-8 +2.4073494341756525e6 7.805241118452836e-8 +2.6281411251997715e6 7.191700112076914e-8 +2.8814215164422425e6 6.600114646022414e-8 +3.173972719651722e6 6.030717489226535e-8 +3.514458156126849e6 5.48375791611718e-8 +3.9140890906038294e6 4.959503748109755e-8 +4.387585118240208e6 4.4582437792327356e-8 +4.9545872114968235e6 3.980290686446132e-8 +5.641785427900708e6 3.525984559488294e-8 +6.486207938885921e6 3.095697234525528e-8 +7.540459571730548e6 2.6898376890302707e-8 +8.881357344415514e6 2.3088588653501798e-8 +1.0624746669295684e7 1.9532664619655167e-8 +1.295214837935377e7 1.62363050698109e-8 +1.6161464064078324e7 1.3206009915200957e-8 +2.0770334439046033e7 1.0449296550255222e-8 +2.774575747970492e7 7.97501535484912e-9 +3.907412538857086e7 5.793829555482476e-9 +5.940504943147255e7 3.918993760367489e-9 +1.0193217517597769e8 2.367735443323899e-9 +2.1812737832284665e8 1.1640586883248531e-9 +8.005144509127394e8 3.4591850987184714e-10 \ No newline at end of file diff --git a/figs/fig-susmag.gplot b/figs/fig-susmag.gplot index c56c898..3140df2 100644 --- a/figs/fig-susmag.gplot +++ b/figs/fig-susmag.gplot @@ -53,7 +53,9 @@ A3 = 1.91747 B3 = B susfunc = "figs/fig-sus_scaling-func.dat" +susfuncm = "figs/fig-sus_scaling-func-mod.dat" magfunc = "figs/fig-mag_scaling-func.dat" +magfuncm = "figs/fig-mag_scaling-func-mod.dat" num = "data/data_square-4096.dat" @@ -64,7 +66,7 @@ set size 1,1 - 0.52 set origin 0,0.52 set xrange [-40:1200] set yrange [-1:33] -set ylabel offset 1,0 '$T_\c\chi|t|^{\gamma}\times 10^3$' +set ylabel offset 1,0 '$T\chi|t|^{\gamma}\times 10^3$' set lmargin 6 set xtics format '' set mxtics 5 @@ -74,9 +76,8 @@ set bmargin 0.2 plot \ num using (X($2, $3)):(10**3 * Tc * $10 * t($2)**gamma):(10**3 * Tc * $11 * t($2)**gamma) with yerrorbars pt 0 lc black, \ susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb cc1 dt 1, \ - susfunc using (10**$1 / B2):(10**(3+$2) * A2 * B2**2 + 10**3 * (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines lw 2 lc rgb cc2 dt 2, \ - susfunc using (10**$1 / B3):(10**(3+$2) * A3 * B3**2 + 10**3 * (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines lw 2 lc rgb cc3 dt 3, \ - -10**3 * (sum[i=1:7] GC(i + 1) * i * (i + 1) * (x)**(i-1)) with lines lw 2 lc rgb cc4 dt 4 + susfuncm using 1:(10**3 * $2) with lines lw 2 lc rgb cc2 dt 2, \ + -10**3 * (sum[i=1:7] GC(i + 1) * i * (i + 1) * (x)**(i-1)) with lines lw 2 lc rgb cc3 dt 3 set bmargin -1 set tmargin 0.2 @@ -90,9 +91,8 @@ set xtics format '%g' plot \ num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)) with yerrorbars pt 0 lc black, \ magfunc using (10**$1 / B):(M0 + 10**($2) * A * B) with linespoints pt 0 lw 2 lc rgb cc1 dt 1, \ - magfunc using (10**$1 / B2):(M0 + 10**($2) * A2 * B2 + (sum[i=1:n] polyint(c(i), lamb, i-1, 10**$1)) / B2) smooth csplines with lines lw 2 lc rgb cc2 dt 2, \ - magfunc using (10**$1 / B3):(M0 + 10**($2) * A3 * B3 + (sum[i=1:n2] polyint(c2(i), lamb2, i-1, 10**$1)) / B3) smooth csplines with lines lw 2 lc rgb cc3 dt 3, \ - -sum[i=1:8] GC(i) * i * x**(i-1) with lines lw 2 lc rgb cc4 dt 4 + magfuncm using 1:2 with lines lw 2 lc rgb cc2 dt 2, \ + -sum[i=1:8] GC(i) * i * x**(i-1) with lines lw 2 lc rgb cc4 dt 3 set logscale xy set tmargin -1 @@ -101,7 +101,7 @@ set size 0.65,0.325 set origin 0.31,0.5 + 0.29 / 2 set xrange [0.0015:1900] set yrange [0.00002:0.08] -set ylabel offset 2.5,0 '\footnotesize$T_\c\chi|t|^\gamma$' +set ylabel offset 2.5,0 '\footnotesize$T\chi|t|^\gamma$' set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\beta\delta}$' set mxtics 5 set xtics format '' -2,10,1000 @@ -112,10 +112,8 @@ set ytics format '\footnotesize$10^{%T}$' 0.00001,10,0.01 plot \ num using (X($2, $3)):(Tc * $10 * t($2)**gamma):(Tc * $11 * t($2)**gamma) with yerrorbars pt 0 lc black, \ susfunc using (10**$1 / B):(10**$2 * A * B**2) with linespoints pt 0 lw 2 lc rgb cc1 dt 1, \ - susfunc using (10**$1 / B2):(10**$2 * A2 * B2**2 + (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines lw 2 lc rgb cc2 dt 2, \ - susfunc using (10**$1 / B3):(10**$2 * A3 * B3**2 + (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines lw 2 lc rgb cc3 dt 3, \ - -sum[i=2:8] GC(i) * i * (i-1) * x**(i-2) with lines dt 4 lw 2 lc rgb cc4, \ - 0.0705991 * x**(-14./15) + susfuncm using 1:2 with lines lw 2 lc rgb cc2 dt 2, \ + -sum[i=2:8] GC(i) * i * (i-1) * x**(i-2) with lines dt 3 lw 2 lc rgb cc4 unset logscale xy set logscale x @@ -129,7 +127,6 @@ set mytics 5 plot \ num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)) with yerrorbars pt 0 lc black, \ magfunc using (10**$1 / B):(M0 + A * B * 10**$2) with linespoints pt 0 lw 2 lc rgb cc1 dt 1, \ - magfunc using (10**$1 / B2):(M0 + 10**($2) * A2 * B2 + (sum[i=1:n] polyint(c(i), lamb, i-1, 10**$1)) / B2) with lines lw 2 lc rgb cc2 dt 2, \ - magfunc using (10**$1 / B3):(M0 + 10**($2) * A3 * B3 + (sum[i=1:n2] polyint(c2(i), lamb2, i-1, 10**$1)) / B3) with lines lw 2 lc rgb cc3 dt 3, \ - -sum[i=1:8] GC(i) * i * x**(i-1) with lines lw 2 lc rgb cc4 dt 4 + magfuncm using 1:2 with lines lw 2 lc rgb cc2 dt 2, \ + -sum[i=1:8] GC(i) * i * x**(i-1) with lines lw 2 lc rgb cc4 dt 3 -- cgit v1.2.3-70-g09d2 From 75153282053738f549f2737f0fe42a18bb17b5a6 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 26 Sep 2017 11:19:54 -0400 Subject: changed paper body to incorporate new analysis --- essential-ising.bib | 13 +++++++++- essential-ising.tex | 67 +++++++++++++++++++++--------------------------- figs/fig-series-data.dat | 16 +++++++----- figs/fig-series.gplot | 10 +++++--- figs/fig-susmag.gplot | 35 +++++++++++++------------ 5 files changed, 74 insertions(+), 67 deletions(-) diff --git a/essential-ising.bib b/essential-ising.bib index 1de2478..64dacb0 100644 --- a/essential-ising.bib +++ b/essential-ising.bib @@ -305,6 +305,18 @@ publisher={IOP Publishing} } +@article{kent-dobias.2018.parametric, + author={Kent-Dobias, Jaron and Sethna, James P}, + journal={(unpublished)}, + year={2018} +} + +@article{kent-dobias.2018.wolff, + author={Kent-Dobias, Jaron and Sethna, James P}, + journal={(unpublished)}, + year={2018} +} + @article{klein.1976.essential, title={Essential singularities at first-order phase transitions}, author={Klein, W and Wallace, DJ and Zia, RKP}, @@ -544,7 +556,6 @@ publisher={APS} } - @article{zinn.1996.universal, title={Universal surface-tension and critical-isotherm amplitude ratios in three dimensions}, author={Zinn, Shun-yong and Fisher, Michael E}, diff --git a/essential-ising.tex b/essential-ising.tex index d641a57..7dd6686 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -316,7 +316,8 @@ the presence of an external field. Briefly, the external field $H$ is applied by adding an extra spin $s_0$ with coupling $|H|$ to all others \cite{dimitrovic.1991.finite}. A quickly converging estimate for the magnetization in the finite-size system was then made by taking -$M=\sgn(H)s_0\sum s_i$, i.e., the magnetization relative to the external spin. +$M=\sgn(H)s_0\sum s_i$, i.e., the magnetization relative to the external spin +\cite{kent-dobias.2018.wolff}. Data was then taken for susceptibility and magnetization for $T_\c-T,H\leq0.1$. This data, rescaled as appropriate to collapse onto a single curve, is plotted in Fig.~\ref{fig:scaling_fits}. @@ -330,34 +331,23 @@ the susceptibility scaling function, then $T\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ and the constant $A$ is fixed to $A=\pi\fX(0)/2B^2=2^{19/8}\pi^3(\arcsinh1)^{15/4}C_0^-$. The resulting scaling functions $\fX$ and $\fM$ are plotted as solid lines in -Fig.~\ref{fig:scaling_fits}. As can be seen, there is very good agreement -between our proposed functional forms and what is measured. However, there -are systematic differences that can be seen most clearly in the magnetization. -Since our method is known to only be accurate for high moments of the free -energy, we should expect that low moments require corrections. Therefore, we -also fit those corrections using -\begin{align} - \tilde\fX^\twodee(X)&=\fX^\twodee(X)+\sum_{n=0}^Nf_n(BX)\label{eq:sus_scaling_poly}\\ - \tilde\fM^\twodee(X)&=\fM^\twodee(X)+\frac{T_\c}B\sum_{n=0}^NF_n(BX)\label{eq:mag_scaling_poly} -\end{align} -where $F_n'(Y)=f_n(Y)$ and -\[ - \begin{aligned} - f_n(Y)&=\frac{c_nx^n}{1+(\lambda Y)^{n+1}}\\ - F_n(Y)&=\frac{c_n\lambda^{-(n+1)}}{n+1}\log(1+(\lambda Y)^{n+1}). - \end{aligned} - \label{eq:poly} -\] -The functions $f_n$ have been chosen to be pure integer power laws for small -argument, but vanish appropriately at large argument. This is necessary -because the susceptibility vanishes with $h|t|^{-\beta\delta}$ while bare -polynomial corrections would not. We fit these functions to known moments of -the free energy's scaling function -\cite{mangazeev.2008.variational,mangazeev.2010.scaling} our numeric data -for $N=0$. The -resulting curves are also plotted as dashed lines in -Fig.~\ref{fig:scaling_fits}. Our singular scaling function with one low-order -correction appears to match data quite well. +Fig.~\ref{fig:scaling_fits}. Though there is good agreement +between our functional forms and what is measured, there +are systematic differences that can be seen most clearly in the +magnetization. This is to be expected based on our earlier discussion: these +scaling forms should only be expected to well-describe the singularity at the +abrupt transition. Our forms both exhibit incorrect low-order +coefficients at the transition (Fig.~\ref{fig:series}) and incorrect +asymptotics as $h|t|^{-\beta\delta}$ becomes very large. In forthcoming work, +we develop a method to incorporate the essential singularity in the scaling +functions into a form that also incorporates known properties of the scaling +functions in the rest of the configuration space using a Schofield-like +parameterization \cite{kent-dobias.2018.parametric}. Fig.~\ref{fig:scaling_fits} shows a result as a +dashed yellow line, which depicts the scaling form resulting from +incorporating our singularity and the known series expansions of the scaling +function at high temperature, low temperature, and at the critical isotherm to +quadratic order. The low-order series coefficients of this modified form are +also shown in Fig.~\ref{fig:series}. \begin{figure} \input{fig-susmag} @@ -370,10 +360,11 @@ correction appears to match data quite well. and $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. The solid blue lines show our analytic results \eqref{eq:sus_scaling} and \eqref{eq:mag_scaling}, the dashed yellow lines show - \eqref{eq:sus_scaling_poly} and \eqref{eq:mag_scaling_poly} for $N=0$, the - dotted green lines show the same for $N=4$, and the red line show the - polynomial resulting from truncating the series after the eight known - terms. + a scaling function modified to match known series expansions + in several known limits, and the + dotted green lines show the + polynomial resulting from truncating the series after the eight terms + reported by \cite{mangazeev.2008.variational,mangazeev.2010.scaling}. } \label{fig:scaling_fits} \end{figure} @@ -383,11 +374,11 @@ correction appears to match data quite well. \caption{ The series coefficients defined by $\tilde\fF(X)=\sum_nf_nX^n$. The blue pluses correspond to the scaling form \eqref{eq:2d_free_scaling}, the - yellow saltires correspond to that form with the first four coefficients - fixed to known values (\eqref{eq:sus_scaling_poly} with $N=0$), the green - stars correspond to that form with the first eight coefficients fixed to - known values (\eqref{eq:sus_scaling_poly} with $N=4$), and the red squares - correspond to the first eight coefficients. + yellow saltires correspond to a scaling function modified to match known + series expansions in several known limits, and the green + stars + correspond to the first eight coefficients from + \cite{mangazeev.2008.variational,mangazeev.2010.scaling}. } \label{fig:series} \end{figure} diff --git a/figs/fig-series-data.dat b/figs/fig-series-data.dat index 8873b25..18e7599 100644 --- a/figs/fig-series-data.dat +++ b/figs/fig-series-data.dat @@ -1,7 +1,9 @@ -0 0.025536974521879204 0.025536974521879207 0.025536974521879207 -1 0.009593512562216971 0.01762155820793826 0.017621558207938266 -2 0.004805345190888446 0.01681987670730995 0.017961054818812806 -3 0.003008718424800099 0.018814565927230777 0.02332905039996879 -4 0.0022605793007909824 0.02318633276267007 0.03634395601353185 -5 0.0019815486405162243 0.0306005114760245 0.06569811399718999 -6 0.001985096734582422 0.04264499878327929 0.13485933959294732 \ No newline at end of file +0 0 0 0 +1 1.22241 1.22241 1.22241 +2 0.012768487260939602 0.012768487260939603 0.012768487260939603 +3 0.0015989187603694952 0.0029369263679897097 0.002936926367989711 +4 0.0004004454325740371 0.0014016563922758293 0.0014967545682344004 +5 0.00015043592124000494 0.0009407282963615389 0.0011664525199984396 +6 0.00007535264335969941 0.000772877758755669 0.0012114652004510617 +7 0.000047179729536100574 0.0007285836065720119 0.0015642408094569044 +8 0.0000354481559746861 0.0007615178354157016 0.002408202492731202 \ No newline at end of file diff --git a/figs/fig-series.gplot b/figs/fig-series.gplot index 171af71..729ff73 100644 --- a/figs/fig-series.gplot +++ b/figs/fig-series.gplot @@ -10,13 +10,15 @@ set logscale y data = "figs/fig-series-data.dat" -set xrange [-0.5:6.5] -set yrange [0.0006:0.4] +set xrange [0.5:8.5] +set yrange [0.000005:5] set key off set xlabel '$n$' -set ylabel offset 2 '$|f_n|$' +set ylabel offset 1 '$|f_n|$' -#set ytics format '\footnotesize$10^{%T}$' 0.001,10,1 +set ytics format '' + +set ytics add ('$\footnotesize10^{-5}$' 10**(-5),'$\footnotesize10^{-4}$' 10**(-4), '$\footnotesize10^{-3}$' 10**(-3),'$\footnotesize10^{-2}$' 10**(-2), '$\footnotesize10^{-1}$' 10**(-1),'$\footnotesize10^{0}$' 10**(0)) plot \ data using 1:2 with points lc rgb cc1, \ diff --git a/figs/fig-susmag.gplot b/figs/fig-susmag.gplot index 3140df2..901576f 100644 --- a/figs/fig-susmag.gplot +++ b/figs/fig-susmag.gplot @@ -75,9 +75,9 @@ set bmargin 0.2 plot \ num using (X($2, $3)):(10**3 * Tc * $10 * t($2)**gamma):(10**3 * Tc * $11 * t($2)**gamma) with yerrorbars pt 0 lc black, \ - susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb cc1 dt 1, \ - susfuncm using 1:(10**3 * $2) with lines lw 2 lc rgb cc2 dt 2, \ - -10**3 * (sum[i=1:7] GC(i + 1) * i * (i + 1) * (x)**(i-1)) with lines lw 2 lc rgb cc3 dt 3 + susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 3 lc rgb cc1 dt 1, \ + susfuncm using 1:(10**3 * $2) with lines lw 3 lc rgb cc2 dt 2, \ + -10**3 * (sum[i=1:7] GC(i + 1) * i * (i + 1) * (x)**(i-1)) with lines lw 3 lc rgb cc3 dt 3 set bmargin -1 set tmargin 0.2 @@ -90,30 +90,31 @@ set xtics format '%g' plot \ num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)) with yerrorbars pt 0 lc black, \ - magfunc using (10**$1 / B):(M0 + 10**($2) * A * B) with linespoints pt 0 lw 2 lc rgb cc1 dt 1, \ - magfuncm using 1:2 with lines lw 2 lc rgb cc2 dt 2, \ - -sum[i=1:8] GC(i) * i * x**(i-1) with lines lw 2 lc rgb cc4 dt 3 + magfunc using (10**$1 / B):(M0 + 10**($2) * A * B) with linespoints pt 0 lw 3 lc rgb cc1 dt 1, \ + magfuncm using 1:2 with lines lw 3 lc rgb cc2 dt 2, \ + -sum[i=1:8] GC(i) * i * x**(i-1) with lines lw 3 lc rgb cc3 dt 3 set logscale xy set tmargin -1 set lmargin -1 -set size 0.65,0.325 -set origin 0.31,0.5 + 0.29 / 2 +set size 0.7,0.325 +set origin 0.26,0.5 + 0.29 / 2 set xrange [0.0015:1900] set yrange [0.00002:0.08] -set ylabel offset 2.5,0 '\footnotesize$T\chi|t|^\gamma$' +set ylabel offset 4,0 '\footnotesize$T\chi|t|^\gamma$' set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\beta\delta}$' set mxtics 5 -set xtics format '' -2,10,1000 +#set xtics format '' -2,10,1000 set xtics add ('$\footnotesize10^{-2}$' 10**(-2), "" 0.1, '$\footnotesize10^0$' 1, "" 10, '$\footnotesize10^2$' 100, "" 1000) set mytics 5 -set ytics format '\footnotesize$10^{%T}$' 0.00001,10,0.01 +#set ytics format '\footnotesize$10^{%T}$' 0.00001,10,0.01 +set ytics add ('$\footnotesize10^{-4}$' 10**(-4), '$\footnotesize10^{-3}$' 10**(-3), '$\footnotesize10^{-2}$' 10**(-2)) plot \ num using (X($2, $3)):(Tc * $10 * t($2)**gamma):(Tc * $11 * t($2)**gamma) with yerrorbars pt 0 lc black, \ - susfunc using (10**$1 / B):(10**$2 * A * B**2) with linespoints pt 0 lw 2 lc rgb cc1 dt 1, \ - susfuncm using 1:2 with lines lw 2 lc rgb cc2 dt 2, \ - -sum[i=2:8] GC(i) * i * (i-1) * x**(i-2) with lines dt 3 lw 2 lc rgb cc4 + susfunc using (10**$1 / B):(10**$2 * A * B**2) with linespoints pt 0 lw 3 lc rgb cc1 dt 1, \ + susfuncm using 1:2 with lines lw 3 lc rgb cc2 dt 2, \ + -sum[i=2:8] GC(i) * i * (i-1) * x**(i-2) with lines dt 3 lw 3 lc rgb cc3 unset logscale xy set logscale x @@ -126,7 +127,7 @@ set mytics 5 plot \ num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)) with yerrorbars pt 0 lc black, \ - magfunc using (10**$1 / B):(M0 + A * B * 10**$2) with linespoints pt 0 lw 2 lc rgb cc1 dt 1, \ - magfuncm using 1:2 with lines lw 2 lc rgb cc2 dt 2, \ - -sum[i=1:8] GC(i) * i * x**(i-1) with lines lw 2 lc rgb cc4 dt 3 + magfunc using (10**$1 / B):(M0 + A * B * 10**$2) with linespoints pt 0 lw 3 lc rgb cc1 dt 1, \ + magfuncm using 1:2 with lines lw 3 lc rgb cc2 dt 2, \ + -sum[i=1:8] GC(i) * i * x**(i-1) with lines lw 3 lc rgb cc3 dt 3 -- cgit v1.2.3-70-g09d2 From af43329a7a7dc6a6270783f3898b931d36e767d3 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 27 Sep 2017 14:30:39 -0400 Subject: new data --- figs/fig-mag_scaling-func-mod.dat | 6642 +++++++----------------------------- figs/fig-series-data.dat | 17 +- figs/fig-sus_scaling-func-mod.dat | 6644 +++++++------------------------------ 3 files changed, 2297 insertions(+), 11006 deletions(-) diff --git a/figs/fig-mag_scaling-func-mod.dat b/figs/fig-mag_scaling-func-mod.dat index 95d94f2..5702933 100644 --- a/figs/fig-mag_scaling-func-mod.dat +++ b/figs/fig-mag_scaling-func-mod.dat @@ -1,5498 +1,1144 @@ -0.0029245327965242416 1.22241 -0.004878604313566581 1.2224597066045628 -0.006836435231669482 1.222509488369378 -0.008798027940567329 1.2225592990482481 -0.010763384834018152 1.2226091385645317 -0.012732508309758007 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1.0628475811150846e-6 +182830.41814630997 8.636320478634407e-7 +235298.25555843086 6.826806573832306e-7 +314759.47254981287 5.205177599506567e-7 +443892.7306092951 3.777837637342556e-7 +675800.0380945331 2.5528527557173693e-7 +1.161212667722158e6 1.540841831223302e-7 +2.4883749168534726e6 7.567867581278825e-8 +9.144911643562693e6 2.2467062041350698e-8 \ No newline at end of file -- cgit v1.2.3-70-g09d2 From 1c392812f20bd2b49cb98cfff5e9445953f7b8fb Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Sat, 30 Sep 2017 20:37:23 -0400 Subject: many changes --- essential-ising.tex | 111 +- figs/fig-mag_scaling-func-mod.dat | 12608 ++++++++++++++++++++++++++++++++---- figs/fig-sus_scaling-func-mod.dat | 11130 +++++++++++++++++++++++++++++-- 3 files changed, 22266 insertions(+), 1583 deletions(-) diff --git a/essential-ising.tex b/essential-ising.tex index 7dd6686..dd2c083 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -32,6 +32,7 @@ % scaling functions \def\fM{\mathcal M} % magnetization \def\fX{\mathcal Y} % susceptibility +\def\fXt{\check{\mathcal Y}} % susceptibility \def\fF{\mathcal F} % free energy \def\fiF{\mathcal H} % imaginary free energy \def\fS{\mathcal S} % surface tension @@ -44,9 +45,9 @@ % dimensions \def\dim{d} -\def\twodee{\textsc{2d} } -\def\threedee{\textsc{3d} } -\def\fourdee{\textsc{4d} } +\def\twodee{\textsc{2d}} +\def\threedee{\textsc{3d}} +\def\fourdee{\textsc{4d}} % fancy partial derivative \newcommand\pd[3][]{ @@ -78,8 +79,9 @@ point. The analytic properties of the free energy are used to determine scaling functions for the free energy in the vicinity of the critical point and the abrupt transition line. These functions have essential singularities - at zero field. Analogous forms for the magnetization and susceptibility in - two dimensions are fit to numeric data and show good agreement. + at the abrupt transition. Analogous forms for the magnetization and susceptibility in + two dimensions are fit to numeric data and show good agreement, especial + when their nonsingular behavior is modified to match existing numeric results. \end{abstract} \maketitle @@ -111,7 +113,7 @@ zero argument, though its effects have long been believed to be unobservable \cite{fisher.1967.condensation}, or simply just neglected \cite{guida.1997.3dising,schofield.1969.parametric,schofield.1969.correlation,caselle.2001.critical,josephson.1969.equation,fisher.1999.trigonometric}. With careful analysis, we have found that assuming the presence of the -essential singularity is predictive of the scaling form of e.g. the +essential singularity is predictive of the scaling form of, for instance, the susceptibility and magnetization. The provenance of the essential singularity can be understood using the @@ -132,20 +134,19 @@ partition function trace to states in the vicinity of the local free energy minimum that characterizes the metastable state. In any case, the free energy develops a nonzero imaginary part in the metastable region. Heuristically, this can be thought of as similar to what happens in quantum mechanics with a -non-unitary Hamiltonian: the imaginary part describes loss of probability in -the system that corresponds to decay. +non-unitary Hamiltonian: the imaginary part describes loss rate of probability +that the system occupies any `accessible' state, which corresponds to decay. In critical droplet theory, the metastable state decays when a domain of the equilibrium state forms whose surface-energy cost for growth is outweighed by bulk-energy gains. There is numerical evidence that, near the critical point, -droplets are spherical \cite{gunther.1993.transfer-matrix}. The free energy +these droplets are spherical \cite{gunther.1993.transfer-matrix}. The free energy cost of the surface of a droplet of radius $R$ is $\Sigma S_\dim R^{\dim-1}$ and that of its bulk is $-M|H|V_\dim R^\dim$, where $S_\dim$ and $V_\dim$ are the surface area and volume of a $(\dim-1)$-sphere, respectively, and $\Sigma$ is the surface tension of the equilibrium--metastable interface. The critical -droplet size then is $R_\c=(\dim-1)\Sigma/M|H|$ and the free energy of the -critical droplet is $\Delta -F_\c=\pi^{\dim/2}\Sigma^\dim((\dim-1)/M|H|)^{\dim-1}/\Gamma(1+\dim/2)$. +droplet size is then $R_\c=(\dim-1)\Sigma/M|H|$ and the free energy of the +critical droplet is $\Delta F_\c=\pi^{\dim/2}\Sigma^\dim((\dim-1)/M|H|)^{\dim-1}/\Gamma(1+\dim/2)$. Assuming the typical singular scaling forms $\Sigma/T=|t|^\mu\fS(h|t|^{-\beta\delta})$ and $M=|t|^\beta\mathcal M(h|t|^{-\beta\delta})$ and using known hyperscaling relations @@ -163,14 +164,15 @@ M(h|t|^{-\beta\delta})$ and using known hyperscaling relations \begin{aligned} \Delta F_\c &=\eqcritformone\\ - &\sim\eqcritformtwo. + &\sim\eqcritformtwo \end{aligned} \] \else \[ - \Delta F_\c=\eqcritformone\sim\eqcritformtwo. + \Delta F_\c=\eqcritformone\sim\eqcritformtwo \] \fi +for the free energy change due to the critical droplet. Since both surface tension and magnetization are finite and nonzero for $H=0$ at $T1$ corresponds to the abrupt transition. In these +coordinates the invariant combination $h|t|^{-\beta\delta}$ is given by +\[ + h|t|^{-\beta\delta}=\frac{h_0g(\theta)}{|1-\theta^2|^{\beta\delta}}=\frac{h_0(-g'(\theta_\c))}{(\theta_c^2-1)^{\beta\delta}}(\theta_\c-\theta) + +\O\big( (\theta_\c-\theta)^2\big), +\] +an analytic function of $\theta$ about $\theta_\c$. +The simplest +function of the coordinate $\theta$ that exhibits the correct singularity at +the abrupt transitions at $h=\pm0$, $t<0$ is +\[ + \fX(\theta)=\fX\bigg(\frac{h_0(-g'(\theta_\c))}{(\theta_\c^2-1)^{\beta\delta}}(\theta_\c-\theta)\bigg)+\fX\bigg(\frac{h_0(-g'(\theta_\c))}{(\theta_\c^2-1)^{\beta\delta}}(\theta_\c+\theta)\bigg) +\] +This function is analytic in the range $-\theta_\c<\theta<\theta_\c$. +In order to correct its low-order behavior to match that expected, we +make use of both the freedom of the coordinate transformation $f$ and an +arbitrary analytic additive function $Y$, +\begin{align} + f(\theta)=\bigg(1-\frac{\theta^2}{\theta_\c^2}\bigg)\sum_{n=0}^\infty + f_n\theta^{2n+1} + && + Y(\theta)=\sum_{n=0}^\infty Y_n\theta^{2n} +\end{align} +so that $\tilde\fX(\theta)=\fX(\theta)+Y(\theta)$. By manipulating thees +coefficients, we can attempt to give the resulting scaling form a series +expansion consistent with known values. One such prediction---made by fixing +the first four terms in the low-temperature, critical isotherm, and +high-temperature expansions +of $\tilde\fX$---is shown as a dashed yellow line in +Fig.~\ref{fig:scaling_fits}. +As shown in Fig.~\ref{fig:series}, the low-order free energy coefficients of this prediction match known values +exactly up to $n=5$, and improve the agreement with higher-order coefficients. + \begin{figure} \input{fig-susmag} diff --git a/figs/fig-mag_scaling-func-mod.dat b/figs/fig-mag_scaling-func-mod.dat index 5702933..657dda5 100644 --- a/figs/fig-mag_scaling-func-mod.dat +++ b/figs/fig-mag_scaling-func-mod.dat @@ -1,1144 +1,11464 @@ -0.009568286124210458 2^(5/16)*ArcSinh[1]^(1/8) -0.014105474133770269 1.222524728271578 -0.018669057029485486 1.2226398161196887 -0.0232591226867679 1.2227552165109419 -0.027875759620704425 1.2228709287034474 -0.032519056989508764 1.2229869519647474 -0.03718910459804977 1.2231032855717547 -0.04188599290129413 1.2232199288105705 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+2.7401057397867832e6 6.91715257237757e-8 +3.033036521041435e6 6.291771757116011e-8 +3.3767982723618587e6 5.692056616680201e-8 +3.7840348717265055e6 5.118363107375861e-8 +4.271616297815629e6 4.57107504091873e-8 +4.8624653768476965e6 4.050608182749496e-8 +5.588381441561218e6 3.5574152817026005e-8 +6.494537437373923e6 3.0919923289751694e-8 +7.646892855190782e6 2.6548864716179547e-8 +9.144911643562693e6 2.2467062041350698e-8 +1.1144434286579737e7 1.8681347803848162e-8 +1.3901197759059438e7 1.51994832335871e-8 +1.785954259148301e7 1.2030410516709167e-8 +2.384947190967022e7 9.184617993503294e-9 +3.357583355491832e7 6.674695388504657e-9 +5.102890777197067e7 4.516234273012404e-9 +8.753053725487643e7 2.729425271488335e-9 +1.8724657831848335e8 1.3422983026281144e-9 +6.86955037774794e8 3.9901028056275807e-10 \ No newline at end of file -- cgit v1.2.3-70-g09d2 From e6d1d9e91a161110d53b019b3e49fbad96dfc396 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Sun, 1 Oct 2017 19:08:30 -0400 Subject: fixed the data --- figs/fig-mag_scaling-func-mod.dat | 12658 +++++++++++++++++++++++++++++++---- figs/fig-series-data.dat | 23 +- figs/fig-sus_scaling-func-mod.dat | 12660 ++++++++++++++++++++++++++++++++---- 3 files changed, 23044 insertions(+), 2297 deletions(-) diff --git a/figs/fig-mag_scaling-func-mod.dat b/figs/fig-mag_scaling-func-mod.dat index 5702933..fc2947a 100644 --- a/figs/fig-mag_scaling-func-mod.dat +++ b/figs/fig-mag_scaling-func-mod.dat @@ -1,1144 +1,11514 @@ -0.009568286124210458 2^(5/16)*ArcSinh[1]^(1/8) -0.014105474133770269 1.222524728271578 -0.018669057029485486 1.2226398161196887 -0.0232591226867679 1.2227552165109419 -0.027875759620704425 1.2228709287034474 -0.032519056989508764 1.2229869519647474 -0.03718910459804977 1.2231032855717547 -0.04188599290129413 1.2232199288105705 -0.046609813007850526 1.2233368809764371 -0.05136065668357925 1.2234541413736797 -0.05613861635507423 1.2235717093156953 -0.06094378511328598 1.2236895841248827 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+4.324844634402942e6 4.518560985022365e-8 +4.970460996897472e6 3.9684288002029715e-8 +5.77637651063881e6 3.449264893050547e-8 +6.801252743816807e6 2.9616805185677633e-8 +8.133548549511681e6 2.506355392160446e-8 +9.91186134264871e6 2.084052654917764e-8 +1.236363151593922e7 1.6956390806482514e-8 +1.588403385982654e7 1.3421132234695928e-8 +2.121123134713153e7 1.0246461659309736e-8 +2.9861427056493342e7 7.446434698516749e-9 +4.538336246018404e7 5.038456459197966e-9 +7.784605934406449e7 3.0450636072710195e-9 +1.66528116960262e8 1.4975395964135683e-9 +6.109401061197972e8 4.4516137129949495e-10 \ No newline at end of file -- cgit v1.2.3-70-g09d2 From 8fae05cab9c8b7699f33dfd1e19a98866d21fd66 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Sun, 1 Oct 2017 19:22:57 -0400 Subject: reordered series data --- figs/fig-series-data.dat | 28 ++++++++++++++-------------- 1 file changed, 14 insertions(+), 14 deletions(-) diff --git a/figs/fig-series-data.dat b/figs/fig-series-data.dat index 238d0c7..4ccd12a 100644 --- a/figs/fig-series-data.dat +++ b/figs/fig-series-data.dat @@ -1,15 +1,15 @@ 1 1.2224099537167044 1.2224099537167044 1.2224099537167044 1.2224099537167044 -2 0.012768487260939603 0.012768487260938698 0.012768487260939602 0.012768487260938698 -3 0.002936926367989711 0.0029369263679897275 0.0015989187603694952 0.0029369263679897275 -4 0.0014967545682344006 0.0014967545682345277 0.0004004454325740371 0.0014967545682345277 -5 0.0011664525199984396 0.001166452519998634 0.00015043592124000494 0.001166452519998634 -6 0.0012114652004510617 0.001176327855808878 0.00007535264335969941 0.001176327855808878 -7 0.0015642408094569044 0.0013967830759039198 0.000047179729536100574 0.0013967830759039198 -8 0.002408202492731202 0.0018511347255072016 0.0000354481559746861 0.0018511347255072016 -9 0 0.0026542024660524974 0.00003107267471478146 0.0026542024660524974 -10 0 0.00403913812864002 0.00003112831239660375 0.00403913812864002 -11 0 0.006442796002996578 0.00003508205591405233 0.006442796002996578 -12 0 0.0106807842579893 0.00004393109082303472 0.0106807842579893 -13 0 0.01829412371722729 0.00006051340948184112 0.01829412371722729 -14 0 0.03224179733419373 0.00009093264412340444 0.03224179733419373 -15 0 0.05830886180596288 0.0001480301306974712 0.05830886180596288 \ No newline at end of file +2 0.012768487260938698 0.012768487260939602 0.012768487260938698 0.012768487260939603 +3 0.0029369263679897275 0.0015989187603694952 0.0029369263679897275 0.002936926367989711 +4 0.0014967545682345277 0.0004004454325740371 0.0014967545682345277 0.0014967545682344006 +5 0.001166452519998634 0.00015043592124000494 0.001166452519998634 0.0011664525199984396 +6 0.001176327855808878 0.00007535264335969941 0.001176327855808878 0.0012114652004510617 +7 0.0013967830759039198 0.000047179729536100574 0.0013967830759039198 0.0015642408094569044 +8 0.0018511347255072016 0.0000354481559746861 0.0018511347255072016 0.002408202492731202 +9 0.0026542024660524974 0.00003107267471478146 0.0026542024660524974 0 +10 0.00403913812864002 0.00003112831239660375 0.00403913812864002 0 +11 0.006442796002996578 0.00003508205591405233 0.006442796002996578 0 +12 0.0106807842579893 0.00004393109082303472 0.0106807842579893 0 +13 0.01829412371722729 0.00006051340948184112 0.01829412371722729 0 +14 0.03224179733419373 0.00009093264412340444 0.03224179733419373 0 +15 0.05830886180596288 0.0001480301306974712 0.05830886180596288 0 \ No newline at end of file -- cgit v1.2.3-70-g09d2 From b96fb40f105e8d23338d274ff12f293c604a6bf5 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 2 Oct 2017 15:52:32 -0400 Subject: final edits before arxiv reupload --- essential-ising.tex | 20 +++++++++++++------- figs/fig-series.gplot | 12 ++++++------ 2 files changed, 19 insertions(+), 13 deletions(-) diff --git a/essential-ising.tex b/essential-ising.tex index dd2c083..f789eb4 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -350,7 +350,7 @@ In forthcoming work, we develop a method to incorporate the essential singularity in the scaling functions into a form that also incorporates known properties of the scaling functions in the rest of the configuration space using a Schofield-like -parameterization \cite{kent-dobias.2018.parametric}. Briefly, we define +parameterization \cite{schofield.1969.parametric,caselle.2001.critical,kent-dobias.2018.parametric}. Briefly, we define parameters $R$ and $\theta$ by \begin{align} t=R(1-\theta^2) @@ -381,7 +381,7 @@ arbitrary analytic additive function $Y$, && Y(\theta)=\sum_{n=0}^\infty Y_n\theta^{2n} \end{align} -so that $\tilde\fX(\theta)=\fX(\theta)+Y(\theta)$. By manipulating thees +so that $\tilde\fX(\theta)=\fX(\theta)+Y(\theta)$. By manipulating these coefficients, we can attempt to give the resulting scaling form a series expansion consistent with known values. One such prediction---made by fixing the first four terms in the low-temperature, critical isotherm, and @@ -390,6 +390,9 @@ of $\tilde\fX$---is shown as a dashed yellow line in Fig.~\ref{fig:scaling_fits}. As shown in Fig.~\ref{fig:series}, the low-order free energy coefficients of this prediction match known values exactly up to $n=5$, and improve the agreement with higher-order coefficients. +Unlike scaling forms which treat $\fX$ as analytic at the coexistence line, +the series coefficients of the scaling form developed here increase without +bound at high order. \begin{figure} @@ -403,10 +406,11 @@ exactly up to $n=5$, and improve the agreement with higher-order coefficients. and $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. The solid blue lines show our analytic results \eqref{eq:sus_scaling} and \eqref{eq:mag_scaling}, the dashed yellow lines show - a scaling function modified to match known series expansions - in several known limits, and the + a scaling function modified to match known series expansions of the + susceptibility + to third order, and the dotted green lines show the - polynomial resulting from truncating the series after the eight terms + polynomial resulting from truncating the known series expansion after the eight terms reported by \cite{mangazeev.2008.variational,mangazeev.2010.scaling}. } \label{fig:scaling_fits} @@ -418,10 +422,12 @@ exactly up to $n=5$, and improve the agreement with higher-order coefficients. The series coefficients defined by $\tilde\fF(X)=\sum_nf_nX^n$. The blue pluses correspond to the scaling form \eqref{eq:2d_free_scaling}, the yellow saltires correspond to a scaling function modified to match known - series expansions in several known limits, and the green + series expansions of the susceptibility to third order---and therefore + the free energy to fifth order---and the green stars correspond to the first eight coefficients from - \cite{mangazeev.2008.variational,mangazeev.2010.scaling}. + \cite{mangazeev.2008.variational,mangazeev.2010.scaling}. The modified + scaling function and the known coefficients match exactly up to $n=5$. } \label{fig:series} \end{figure} diff --git a/figs/fig-series.gplot b/figs/fig-series.gplot index 729ff73..49aa4cc 100644 --- a/figs/fig-series.gplot +++ b/figs/fig-series.gplot @@ -1,5 +1,5 @@ -set terminal pslatex rotate size 3.417,2.111 +set terminal pslatex rotate size 3.6,2.111 cc1 = "#5e81b5" cc2 = "#e19c24" @@ -10,18 +10,18 @@ set logscale y data = "figs/fig-series-data.dat" -set xrange [0.5:8.5] +set xrange [0.5:15.5] set yrange [0.000005:5] set key off set xlabel '$n$' -set ylabel offset 1 '$|f_n|$' +set ylabel offset 2 '$|f_n|$' set ytics format '' set ytics add ('$\footnotesize10^{-5}$' 10**(-5),'$\footnotesize10^{-4}$' 10**(-4), '$\footnotesize10^{-3}$' 10**(-3),'$\footnotesize10^{-2}$' 10**(-2), '$\footnotesize10^{-1}$' 10**(-1),'$\footnotesize10^{0}$' 10**(0)) plot \ - data using 1:2 with points lc rgb cc1, \ - data using 1:3 with points lc rgb cc2, \ - data using 1:4 with points lc rgb cc3 + data using 1:3 with points lc rgb cc1, \ + data using 1:2 with points lc rgb cc2, \ + data using 1:5 with points lc rgb cc3 -- cgit v1.2.3-70-g09d2 From 224b34bb3240fe28b04ddcb298439531f06e738c Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 2 Oct 2017 15:59:04 -0400 Subject: fixed a spelling mistake --- essential-ising.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/essential-ising.tex b/essential-ising.tex index f789eb4..33f6fe9 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -80,7 +80,7 @@ scaling functions for the free energy in the vicinity of the critical point and the abrupt transition line. These functions have essential singularities at the abrupt transition. Analogous forms for the magnetization and susceptibility in - two dimensions are fit to numeric data and show good agreement, especial + two dimensions are fit to numeric data and show good agreement, especially when their nonsingular behavior is modified to match existing numeric results. \end{abstract} -- cgit v1.2.3-70-g09d2 From 01f16c5f0288bec93c6c2a97780dc8e1e0723cde Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 2 Oct 2017 16:07:07 -0400 Subject: rejiggered graphs --- figs/fig-mag_scaling-func-mod.dat | 2 +- figs/fig-series.gplot | 4 +++- figs/fig-sus_scaling-func-mod.dat | 2 +- 3 files changed, 5 insertions(+), 3 deletions(-) diff --git a/figs/fig-mag_scaling-func-mod.dat b/figs/fig-mag_scaling-func-mod.dat index 8a6e373..fc2947a 100644 --- a/figs/fig-mag_scaling-func-mod.dat +++ b/figs/fig-mag_scaling-func-mod.dat @@ -11511,4 +11511,4 @@ 3.762239475833869e7 3.3373472708378333 6.161471090212426e7 3.434242107873912 1.2218708815216324e8 3.5654485387101857 -3.8873411154002964e8 3.766872238228526 +3.8873411154002964e8 3.766872238228526 \ No newline at end of file diff --git a/figs/fig-series.gplot b/figs/fig-series.gplot index 49aa4cc..be3876b 100644 --- a/figs/fig-series.gplot +++ b/figs/fig-series.gplot @@ -1,5 +1,5 @@ -set terminal pslatex rotate size 3.6,2.111 +set terminal pslatex rotate size 3.417,2.111 cc1 = "#5e81b5" cc2 = "#e19c24" @@ -16,6 +16,8 @@ set key off set xlabel '$n$' set ylabel offset 2 '$|f_n|$' +set lmargin 7 + set ytics format '' set ytics add ('$\footnotesize10^{-5}$' 10**(-5),'$\footnotesize10^{-4}$' 10**(-4), '$\footnotesize10^{-3}$' 10**(-3),'$\footnotesize10^{-2}$' 10**(-2), '$\footnotesize10^{-1}$' 10**(-1),'$\footnotesize10^{0}$' 10**(0)) diff --git a/figs/fig-sus_scaling-func-mod.dat b/figs/fig-sus_scaling-func-mod.dat index a1035a2..5d49f9b 100644 --- a/figs/fig-sus_scaling-func-mod.dat +++ b/figs/fig-sus_scaling-func-mod.dat @@ -11512,4 +11512,4 @@ 4.538336246018404e7 5.038456459197966e-9 7.784605934406449e7 3.0450636072710195e-9 1.66528116960262e8 1.4975395964135683e-9 -6.109401061197972e8 4.4516137129949495e-10 +6.109401061197972e8 4.4516137129949495e-10 \ No newline at end of file -- cgit v1.2.3-70-g09d2 From 883368ecef475949b961955f5857ff7ab3944d0d Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 2 Oct 2017 16:34:05 -0400 Subject: changed f to g in some places --- essential-ising.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/essential-ising.tex b/essential-ising.tex index 33f6fe9..e125229 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -373,11 +373,11 @@ the abrupt transitions at $h=\pm0$, $t<0$ is \] This function is analytic in the range $-\theta_\c<\theta<\theta_\c$. In order to correct its low-order behavior to match that expected, we -make use of both the freedom of the coordinate transformation $f$ and an +make use of both the freedom of the coordinate transformation $g$ and an arbitrary analytic additive function $Y$, \begin{align} - f(\theta)=\bigg(1-\frac{\theta^2}{\theta_\c^2}\bigg)\sum_{n=0}^\infty - f_n\theta^{2n+1} + g(\theta)=\bigg(1-\frac{\theta^2}{\theta_\c^2}\bigg)\sum_{n=0}^\infty + g_n\theta^{2n+1} && Y(\theta)=\sum_{n=0}^\infty Y_n\theta^{2n} \end{align} -- cgit v1.2.3-70-g09d2 From e11ae4bdaacb3c3e386039ab013ce1ed514b9792 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 22 Dec 2020 19:41:25 +0100 Subject: Deleted a lot. --- .gitignore | 5 +- essential-ising.bib | 570 -- essential-ising_prl.txt | 18 - essential-ising_prl2.txt | 1 - figs/fig-mag_scaling-func-mod.dat | 11514 ----------------------------------- figs/fig-mag_scaling-func.dat | 10001 ------------------------------- figs/fig-series-data.dat | 15 - figs/fig-series.gplot | 29 - figs/fig-sus_scaling-func-mod.dat | 11515 ------------------------------------ figs/fig-sus_scaling-func.dat | 10001 ------------------------------- figs/fig-susmag.gplot | 133 - makefile | 21 - refs/PhysRevB.40.9006.pdf | Bin 670094 -> 0 bytes 13 files changed, 1 insertion(+), 43822 deletions(-) delete mode 100644 essential-ising.bib delete mode 100644 essential-ising_prl.txt delete mode 100644 essential-ising_prl2.txt delete mode 100644 figs/fig-mag_scaling-func-mod.dat delete mode 100644 figs/fig-mag_scaling-func.dat delete mode 100644 figs/fig-series-data.dat delete mode 100644 figs/fig-series.gplot delete mode 100644 figs/fig-sus_scaling-func-mod.dat delete mode 100644 figs/fig-sus_scaling-func.dat delete mode 100644 figs/fig-susmag.gplot delete mode 100644 makefile delete mode 100644 refs/PhysRevB.40.9006.pdf diff --git a/.gitignore b/.gitignore index e413095..034b355 100644 --- a/.gitignore +++ b/.gitignore @@ -4,7 +4,4 @@ *.blg *Notes.bib *.dvi -essential-ising.pdf - -figs/*.tex -fig-*.tex +/*.pdf diff --git a/essential-ising.bib b/essential-ising.bib deleted file mode 100644 index 64dacb0..0000000 --- a/essential-ising.bib +++ /dev/null @@ -1,570 +0,0 @@ - -@article{ambegaokar.1978.dissipation, - author = "V. 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II. Phase-space cell analysis of critical behavior}, - author={Wilson, Kenneth G}, - journal={Physical Review B}, - volume={4}, - number={9}, - pages={3184}, - year={1971}, - publisher={APS} -} - -@article{wu.1976.spin, - title={Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region}, - author={Wu, Tai Tsun and McCoy, Barry M and Tracy, Craig A and Barouch, Eytan}, - journal={Physical Review B}, - volume={13}, - number={1}, - pages={316}, - year={1976}, - publisher={APS} -} - -@article{zinn.1996.universal, - title={Universal surface-tension and critical-isotherm amplitude ratios in three dimensions}, - author={Zinn, Shun-yong and Fisher, Michael E}, - journal={Physica A: Statistical Mechanics and its Applications}, - volume={226}, - number={1-2}, - pages={168--180}, - year={1996}, - publisher={Elsevier} -} - - diff --git a/essential-ising_prl.txt b/essential-ising_prl.txt deleted file mode 100644 index 41039c1..0000000 --- a/essential-ising_prl.txt +++ /dev/null @@ -1,18 +0,0 @@ - -The renormalization group and the scaling theories that result from its use -have become essential to our understanding of singular behavior in -thermodynamic functions at continuous phase transitions. Despite this, they -have long been viewed as irrelevant to the quantitative study of abrupt phase -transitions, and likewise the presence of an abrupt phase transition has -typically been considered irrelevant to scaling analysis. Our work -demonstrates, for one common model with both a continuous and a -line of abrupt transitions, that well-known singularities in thermodynamic -functions at the abrupt transition long assumed to be unobservable are in fact -part of the scaling theory of the critical point, and make explicit -predictions confirmed by numerics. This result indicates that many published -statistical models whose abrupt transitions were not incorporated into scaling -theories should be revisited, and introduces a novel new method for -determining new universal scaling properties in previously poorly-understood -regimes. - - diff --git a/essential-ising_prl2.txt b/essential-ising_prl2.txt deleted file mode 100644 index 1cc1f44..0000000 --- a/essential-ising_prl2.txt +++ /dev/null @@ -1 +0,0 @@ -The Ising model critical point governs the termination of a line of abrupt transitions (e.g., the liquid-gas transition endpoint at high pressure). The universal scaling functions describing the critical behavior in non-zero field have long been a challenge. We show that these scaling functions inherit an essential singularity from the abrupt transition line, associated with the nucleation of droplets in the metastable phase (e.g. the supercooled gas), giving an analytic formula for the singularity and fitting to simulations. This same basic method may be useful in studying other critical points with metastability, such as in glasses and disordered systems. diff --git a/figs/fig-mag_scaling-func-mod.dat b/figs/fig-mag_scaling-func-mod.dat deleted file mode 100644 index fc2947a..0000000 --- a/figs/fig-mag_scaling-func-mod.dat +++ /dev/null @@ -1,11514 +0,0 @@ -0.0007052661693227589 1.222409953716735 -0.001122417188767094 1.222420598483419 -0.0015398054716768543 1.222431246058725 -0.001957431099433065 1.2224418966267505 -0.0023752941534654344 1.2224525501764913 -0.0027933947152594856 1.2224632067121646 -0.00321173286636588 1.2224738662400279 -0.0036303086883853297 1.2224845287591373 -0.00404912226296338 1.2224951942680957 -0.004468173671817067 1.222505862767504 -0.004887462996716529 1.2225165342554214 -0.005306990319480478 1.2225272087320074 -0.00572675572197753 1.2225378861967173 -0.006146759286147286 1.2225485666489038 -0.006567001093987319 1.2225592500879203 -0.006987481227529069 1.222569936513495 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-4.240178736365 --4.043 -4.239178868334 --4.042 -4.238179004978 --4.041 -4.237179141937 --4.04 -4.236179279202 --4.039 -4.235179416779 --4.038 -4.234179554673 --4.037 -4.233179688575 --4.036 -4.232179827102 --4.035 -4.231179965947 --4.034 -4.230180105104 --4.033 -4.229180244581 --4.032 -4.228180384371 --4.031 -4.227180524477 --4.03 -4.226180660676 --4.029 -4.22518080143 --4.0280000000000005 -4.224180942508 --4.027 -4.223181083903 --4.026 -4.222181225619 --4.025 -4.221181367662 --4.024 -4.220181505849 --4.023 -4.219181648543 --4.022 -4.218181791566 --4.021 -4.21718193491 --4.02 -4.216182078579 --4.019 -4.21518222258 --4.018 -4.214182366904 --4.017 -4.213182507453 --4.016 -4.212182652444 --4.015 -4.211182797764 --4.014 -4.210182943419 --4.013 -4.209183089401 --4.0120000000000005 -4.208183235718 --4.011 -4.207183378315 --4.01 -4.206183525305 --4.009 -4.205183672633 --4.008 -4.204183820292 --4.007 -4.203183968287 --4.006 -4.202184116624 --4.005 -4.201184265294 --4.004 -4.200184410315 --4.003 -4.199184559676 --4.002 -4.198184709372 --4.001 -4.197184859413 --4. -4.196185009791 --3.9989999999999997 -4.195185160512 --3.998 -4.194185307648 --3.997 -4.193185459065 --3.996 -4.192185610829 --3.995 -4.191185762939 --3.9939999999999998 -4.190185915391 --3.993 -4.189186068194 --3.992 -4.188186217467 --3.9909999999999997 -4.187186370973 --3.99 -4.186186524834 --3.989 -4.185186679039 --3.988 -4.184186833599 --3.987 -4.183186988511 --3.9859999999999998 -4.182187143773 --3.985 -4.181187295576 --3.984 -4.180187451557 --3.9829999999999997 -4.179187607892 --3.982 -4.178187764587 --3.981 -4.177187921635 --3.98 -4.176188079042 --3.979 -4.175188233046 --3.9779999999999998 -4.174188391178 --3.977 -4.173188549674 --3.976 -4.17218870853 --3.975 -4.171188867748 --3.974 -4.170189027329 --3.973 -4.169189187272 --3.972 -4.168189343879 --3.971 -4.167189504563 --3.9699999999999998 -4.16618966561 --3.9690000000000003 -4.16518982703 --3.968 -4.164189988813 --3.967 -4.163190150967 --3.966 -4.162190309839 --3.965 -4.161190472739 --3.964 -4.160190636013 --3.963 -4.159190799659 --3.9619999999999997 -4.158190963677 --3.9610000000000003 -4.157191128073 --3.96 -4.156191292839 --3.959 -4.155191454391 --3.958 -4.15419161992 --3.957 -4.153191785825 --3.956 -4.152191952111 --3.955 -4.151192118776 --3.9539999999999997 -4.150192285819 --3.9530000000000003 -4.149192449703 --3.952 -4.148192617517 --3.951 -4.147192785717 --3.95 -4.1461929543 --3.949 -4.145193123266 --3.948 -4.144193292619 --3.947 -4.143193462359 --3.9459999999999997 -4.142193628997 --3.9450000000000003 -4.141193799522 --3.944 -4.140193970433 --3.943 -4.139194141736 --3.942 -4.13819431343 --3.941 -4.137194485514 --3.94 -4.136194654553 --3.939 -4.135194827433 --3.9379999999999997 -4.134195000706 --3.9370000000000003 -4.133195174379 --3.936 -4.132195348445 --3.935 -4.131195522909 --3.934 -4.130195697772 --3.933 -4.129195869648 --3.932 -4.128196045318 --3.931 -4.12719622139 --3.9299999999999997 -4.126196397861 --3.9290000000000003 -4.12519657474 --3.928 -4.124196752019 --3.927 -4.123196926365 --3.926 -4.122197104465 --3.925 -4.121197282969 --3.924 -4.120197461883 --3.923 -4.119197641205 --3.9219999999999997 -4.118197820935 --3.9210000000000003 -4.11719800108 --3.92 -4.116198178347 --3.919 -4.115198359321 --3.918 -4.114198540709 --3.917 -4.113198722509 --3.916 -4.112198904727 --3.915 -4.111199087361 --3.9139999999999997 -4.110199267171 --3.9130000000000003 -4.10919945065 --3.912 -4.108199634545 --3.911 -4.107199818861 --3.91 -4.106200003599 --3.909 -4.105200187156 --3.908 -4.104200372743 --3.907 -4.103200558755 --3.9059999999999997 -4.102200743599 --3.9050000000000002 -4.101200930467 --3.904 -4.100201117758 --3.903 -4.099201303901 --3.902 -4.098201492056 --3.901 -4.097201680639 --3.9 -4.096201869654 --3.899 -4.095202057537 --3.8979999999999997 -4.09420224742 --3.8970000000000002 -4.093202437741 --3.896 -4.092202626939 --3.895 -4.091202818133 --3.894 -4.090203009764 --3.893 -4.089203200288 --3.892 -4.0882033928 --3.891 -4.087203585753 --3.8899999999999997 -4.086203777612 --3.8890000000000002 -4.085203971453 --3.888 -4.084204165734 --3.887 -4.083204358939 --3.886 -4.082204554115 --3.885 -4.081204749736 --3.884 -4.080204945806 --3.883 -4.079205140817 --3.8819999999999997 -4.078205337787 --3.8810000000000002 -4.077205535211 --3.88 -4.076205731585 --3.879 -4.075205929915 --3.878 -4.074206128699 --3.877 -4.073206326449 --3.876 -4.072206526147 --3.875 -4.071206726302 --3.8739999999999997 -4.070206925436 --3.873 -4.069207126513 --3.872 -4.068207328047 --3.871 -4.067207530044 --3.87 -4.066207731039 --3.8689999999999998 -4.065207933963 --3.868 -4.064208137353 --3.867 -4.063208339756 --3.8659999999999997 -4.062208544079 --3.865 -4.061208748874 --3.864 -4.060208952691 --3.863 -4.059209158426 --3.862 -4.058209364632 --3.8609999999999998 -4.057209569875 --3.86 -4.05620977703 --3.859 -4.055209984659 --3.8579999999999997 -4.054210192762 --3.857 -4.053210399923 --3.856 -4.052210608983 --3.855 -4.051210818523 --3.854 -4.050211027132 --3.8529999999999998 -4.049211237634 --3.852 -4.04821144862 --3.851 -4.047211658688 --3.8499999999999996 -4.046211870643 --3.849 -4.045212083086 --3.848 -4.044212294622 --3.847 -4.043212508041 --3.846 -4.042212721949 --3.8449999999999998 -4.041212936345 --3.8440000000000003 -4.040213149857 --3.843 -4.03921336524 --3.842 -4.038213581117 --3.841 -4.037213796119 --3.84 -4.036214012988 --3.839 -4.035214230356 --3.838 -4.034214446861 --3.8369999999999997 -4.033214665228 --3.8360000000000003 -4.032214884097 --3.835 -4.031215102114 --3.834 -4.030215321988 --3.833 -4.029215542367 --3.832 -4.028215761907 --3.831 -4.0272159833 --3.83 -4.026216205199 --3.8289999999999997 -4.025216427605 --3.8280000000000003 -4.024216649195 --3.827 -4.023216872625 --3.826 -4.022217096568 --3.825 -4.021217319705 --3.824 -4.020217544677 --3.823 -4.019217770166 --3.822 -4.018217994861 --3.8209999999999997 -4.017218221387 --3.8200000000000003 -4.016218448434 --3.819 -4.015218674697 --3.818 -4.014218902788 --3.817 -4.013219131402 --3.816 -4.012219360537 --3.815 -4.01121958891 --3.814 -4.010219819102 --3.8129999999999997 -4.00922004982 --3.8120000000000003 -4.008220279789 --3.811 -4.007220511569 --3.81 -4.006220743881 --3.809 -4.005220975454 --3.808 -4.004221208834 --3.807 -4.003221442751 --3.806 -4.00222167594 --3.8049999999999997 -4.001221910932 --3.8040000000000003 -4.000222146465 --3.803 -3.999222382536 --3.802 -3.998222617897 --3.801 -3.997222855055 --3.8 -3.996223092756 --3.799 -3.99522332976 --3.798 -3.994223568556 --3.7969999999999997 -3.993223807898 --3.7960000000000003 -3.992224046557 --3.795 -3.991224287001 --3.794 -3.990224527997 --3.793 -3.98922476832 --3.792 -3.988225010425 --3.791 -3.987225253086 --3.79 -3.986225495086 --3.7889999999999997 -3.985225738862 --3.7880000000000003 -3.9842259832 --3.787 -3.983226228096 --3.786 -3.982226472349 --3.785 -3.981226718373 --3.784 -3.98022696496 --3.783 -3.979227210917 --3.782 -3.97822745864 --3.7809999999999997 -3.977227706931 --3.7800000000000002 -3.976227954604 --3.779 -3.975228204038 --3.778 -3.974228454044 --3.777 -3.973228703443 --3.776 -3.9722289546 --3.775 -3.971229206334 --3.774 -3.970229458645 --3.7729999999999997 -3.969229710363 --3.7720000000000002 -3.968229963838 --3.771 -3.967230217891 --3.77 -3.966230471365 --3.769 -3.965230726589 --3.768 -3.964230982397 --3.767 -3.963231237639 --3.766 -3.962231494626 --3.7649999999999997 -3.961231752202 --3.7640000000000002 -3.960232009224 --3.763 -3.959232267985 --3.762 -3.958232527342 --3.761 -3.957232787293 --3.76 -3.956233046703 --3.759 -3.955233307851 --3.758 -3.954233569599 --3.7569999999999997 -3.953233830817 --3.7560000000000002 -3.952234093771 --3.755 -3.951234357325 --3.754 -3.950234620365 --3.753 -3.949234885134 --3.752 -3.94823515051 --3.751 -3.947235415382 --3.75 -3.946235681981 --3.7489999999999997 -3.945235949191 --3.748 -3.94423621591 --3.747 -3.94323648435 --3.746 -3.942236753407 --3.745 -3.941237023081 --3.7439999999999998 -3.940237292278 --3.743 -3.939237563194 --3.742 -3.938237834732 --3.7409999999999997 -3.937238105804 --3.74 -3.936238378593 --3.739 -3.935238652006 --3.738 -3.934238924968 --3.737 -3.933239199641 --3.7359999999999998 -3.932239474944 --3.735 -3.931239749807 --3.734 -3.930240026378 --3.7329999999999997 -3.929240303583 --3.732 -3.928240581427 --3.731 -3.927240858843 --3.73 -3.926241137965 --3.729 -3.925241417728 --3.7279999999999998 -3.924241697076 --3.727 -3.923241978127 --3.726 -3.922242259823 --3.7249999999999996 -3.921242541116 --3.724 -3.920242824111 --3.723 -3.919243107753 --3.722 -3.918243391006 --3.721 -3.917243675955 --3.7199999999999998 -3.916243961557 --3.7190000000000003 -3.915244247818 --3.718 -3.914244533701 --3.717 -3.913244821279 --3.716 -3.912245109517 --3.715 -3.91124539739 --3.714 -3.910245686955 --3.713 -3.909245977186 --3.7119999999999997 -3.908246267063 --3.7110000000000003 -3.907246558631 --3.71 -3.906246850866 --3.709 -3.905247142762 --3.708 -3.904247436344 --3.707 -3.903247730599 --3.706 -3.902248025531 --3.705 -3.901248320139 --3.7039999999999997 -3.900248616428 --3.7030000000000003 -3.899248913399 --3.702 -3.898249210056 --3.701 -3.897249508394 --3.7 -3.896249807418 --3.699 -3.895250106139 --3.698 -3.89425040654 --3.697 -3.89325070763 --3.6959999999999997 -3.89225100843 --3.6950000000000003 -3.891251310908 --3.694 -3.890251614079 --3.693 -3.889251916973 --3.692 -3.888252221541 --3.691 -3.887252526807 --3.69 -3.886252832776 --3.689 -3.885253138482 --3.6879999999999997 -3.88425344586 --3.6870000000000003 -3.883253753944 --3.686 -3.882254061777 --3.685 -3.881254371279 --3.684 -3.880254681493 --3.683 -3.879254991467 --3.682 -3.878255303109 --3.681 -3.877255615468 --3.6799999999999997 -3.876255927597 --3.6790000000000003 -3.875256241396 --3.678 -3.874256555913 --3.677 -3.873256871153 --3.676 -3.872257186181 --3.675 -3.871257502872 --3.674 -3.870257820292 --3.673 -3.869258137512 --3.6719999999999997 -3.868258456392 --3.6710000000000003 -3.867258776008 --3.67 -3.866259095432 --3.669 -3.865259416519 --3.668 -3.864259738343 --3.667 -3.863260059989 --3.666 -3.862260383296 --3.665 -3.861260707345 --3.6639999999999997 -3.860261032138 --3.6630000000000003 -3.85926135677 --3.662 -3.85826168306 --3.661 -3.857262010099 --3.66 -3.856262336987 --3.659 -3.855262665533 --3.658 -3.854262994834 --3.657 -3.853263323995 --3.6559999999999997 -3.852263654812 --3.6550000000000002 -3.851263986391 --3.654 -3.85026431784 --3.653 -3.849264650945 --3.652 -3.848264984816 --3.651 -3.847265318569 --3.65 -3.846265653977 --3.649 -3.845265990157 --3.6479999999999997 -3.844266327108 --3.6470000000000002 -3.843266663959 --3.646 -3.842267002462 --3.645 -3.841267341744 --3.644 -3.840267680935 --3.643 -3.839268021779 --3.642 -3.838268363407 --3.641 -3.837268704956 --3.6399999999999997 -3.836269048157 --3.6390000000000002 -3.835269392149 --3.638 -3.83426973607 --3.637 -3.833270081645 --3.636 -3.832270428015 --3.635 -3.831270775179 --3.634 -3.830271122292 --3.633 -3.829271471056 --3.6319999999999997 -3.828271820621 --3.6310000000000002 -3.827272170147 --3.63 -3.826272521323 --3.629 -3.825272873306 --3.628 -3.82427322526 --3.627 -3.823273578864 --3.626 -3.822273933281 --3.625 -3.82127428768 --3.6239999999999997 -3.82027464373 --3.623 -3.8192750006 --3.622 -3.818275358287 --3.621 -3.817275715973 --3.62 -3.816276075309 --3.6189999999999998 -3.815276435471 --3.618 -3.814276795642 --3.617 -3.813277157463 --3.6159999999999997 -3.812277520116 --3.615 -3.81127788279 --3.614 -3.810278247113 --3.613 -3.809278612274 --3.612 -3.808278977467 --3.6109999999999998 -3.807279344309 --3.61 -3.806279711996 --3.609 -3.805280080127 --3.6079999999999997 -3.804280449105 --3.607 -3.803280819336 --3.606 -3.802281190017 --3.605 -3.801281561952 --3.604 -3.800281934345 --3.6029999999999998 -3.799282307595 --3.602 -3.798282682102 --3.601 -3.797283057076 --3.5999999999999996 -3.796283432914 --3.599 -3.795283810012 --3.598 -3.794284187583 --3.597 -3.793284566417 --3.596 -3.792284945732 --3.5949999999999998 -3.791285325921 --3.594 -3.790285707374 --3.593 -3.789286089319 --3.592 -3.788286472143 --3.591 -3.787286856235 --3.59 -3.786287240825 --3.589 -3.785287626686 --3.588 -3.784288013051 --3.5869999999999997 -3.783288400307 --3.5860000000000003 -3.782288788838 --3.585 -3.78128917788 --3.584 -3.780289568199 --3.583 -3.779289959037 --3.582 -3.778290350775 --3.581 -3.777290743794 --3.58 -3.776291137341 --3.5789999999999997 -3.775291531794 --3.5780000000000003 -3.774291927532 --3.577 -3.773292323805 --3.576 -3.772292721365 --3.575 -3.771293119467 --3.574 -3.770293518487 --3.573 -3.769293918796 --3.572 -3.768294319657 --3.5709999999999997 -3.767294721442 --3.5700000000000003 -3.766295124521 --3.569 -3.765295528159 --3.568 -3.764295933094 --3.567 -3.763296338595 --3.566 -3.76229674503 --3.565 -3.761297152766 --3.564 -3.760297561077 --3.5629999999999997 -3.759297970689 --3.5620000000000003 -3.758298380885 --3.561 -3.757298792025 --3.56 -3.756299204472 --3.559 -3.755299617509 --3.558 -3.754300031498 --3.557 -3.753300446797 --3.556 -3.752300862697 --3.5549999999999997 -3.751301279908 --3.5540000000000003 -3.750301697727 --3.553 -3.749302116507 --3.552 -3.748302536606 --3.551 -3.747302957319 --3.55 -3.746303379001 --3.549 -3.745303802005 --3.548 -3.744304225633 --3.5469999999999997 -3.743304650585 --3.5460000000000003 -3.742305076168 --3.545 -3.741305502731 --3.544 -3.740305930623 --3.543 -3.739306359154 --3.542 -3.738306789016 --3.541 -3.737307219524 --3.54 -3.736307651024 --3.5389999999999997 -3.73530808386 --3.5380000000000003 -3.734308517351 --3.537 -3.73330895184 --3.536 -3.732309387671 --3.535 -3.731309824165 --3.534 -3.730310262002 --3.533 -3.729310700509 --3.532 -3.728311140027 --3.5309999999999997 -3.727311580893 --3.5300000000000002 -3.72631202244 --3.529 -3.725312465002 --3.528 -3.724312908918 --3.527 -3.723313353523 --3.526 -3.722313799483 --3.525 -3.721314246139 --3.524 -3.720314693823 --3.5229999999999997 -3.719315142869 --3.5220000000000002 -3.718315592619 --3.521 -3.717316043733 --3.52 -3.716316495557 --3.519 -3.715316948423 --3.518 -3.714317402657 --3.517 -3.713317857611 --3.516 -3.712318313614 --3.5149999999999997 -3.711318770991 --3.5140000000000002 -3.710319229097 --3.513 -3.70931968858 --3.512 -3.7083201488 --3.511 -3.707320610079 --3.51 -3.706321072741 --3.509 -3.705321536148 --3.508 -3.704322000623 --3.5069999999999997 -3.703322466485 --3.5060000000000002 -3.702322933104 --3.505 -3.701323401112 --3.504 -3.700323869881 --3.503 -3.699324339731 --3.502 -3.698324810976 --3.501 -3.697325282994 --3.5 -3.69632575641 --3.4989999999999997 -3.695326230603 --3.498 -3.694326705891 --3.497 -3.693327182582 --3.496 -3.69232766006 --3.495 -3.691328138639 --3.4939999999999998 -3.690328618628 --3.493 -3.689329099413 --3.492 -3.688329581614 --3.4909999999999997 -3.687330064616 --3.49 -3.686330548733 --3.489 -3.685331034269 --3.488 -3.684331520618 --3.487 -3.68333200839 --3.4859999999999998 -3.682332496981 --3.485 -3.681332986699 --3.484 -3.680333477846 --3.4829999999999997 -3.679333969821 --3.482 -3.678334462932 --3.481 -3.677334957475 --3.48 -3.676335452859 --3.479 -3.675335949682 --3.4779999999999998 -3.674336447349 --3.477 -3.673336946164 --3.476 -3.672337446424 --3.4749999999999996 -3.671337947538 --3.474 -3.670338449808 --3.473 -3.669338953528 --3.472 -3.668339458114 --3.471 -3.667339964154 --3.4699999999999998 -3.666340471066 --3.469 -3.665340979147 --3.468 -3.664341488688 --3.4669999999999996 -3.663341999111 --3.466 -3.662342511 --3.465 -3.661343023777 --3.464 -3.660343537736 --3.463 -3.659344053165 --3.4619999999999997 -3.658344569493 --3.4610000000000003 -3.657345087012 --3.46 -3.656345606006 --3.459 -3.655346125911 --3.458 -3.654346647297 --3.457 -3.653347169598 --3.456 -3.652347693103 --3.455 -3.651348218096 --3.4539999999999997 -3.650348744014 --3.4530000000000003 -3.649349271146 --3.452 -3.64834979977 --3.451 -3.647350329331 --3.45 -3.64635086039 --3.449 -3.645351392393 --3.448 -3.644351925621 --3.447 -3.643352460355 --3.4459999999999997 -3.642352996041 --3.4450000000000003 -3.641353533238 --3.444 -3.640354071395 --3.443 -3.639354610792 --3.442 -3.638355151704 --3.441 -3.637355693588 --3.44 -3.63635623672 --3.439 -3.635356781375 --3.4379999999999997 -3.634357327011 --3.4370000000000003 -3.633357874175 --3.436 -3.632358422327 --3.435 -3.631358971742 --3.434 -3.63035952269 --3.433 -3.629360074638 --3.432 -3.628360627857 --3.431 -3.627361182617 --3.4299999999999997 -3.626361738386 --3.4290000000000003 -3.625362295702 --3.428 -3.624362854034 --3.427 -3.623363413653 --3.426 -3.622363974823 --3.425 -3.621364537022 --3.424 -3.620365100777 --3.423 -3.619365665568 --3.4219999999999997 -3.618366231659 --3.4210000000000003 -3.617366799315 --3.42 -3.616367368016 --3.419 -3.615367938027 --3.418 -3.614368509609 --3.417 -3.613369082248 --3.416 -3.612369656463 --3.415 -3.611370231742 --3.4139999999999997 -3.610370808346 --3.4130000000000003 -3.609371386534 --3.412 -3.608371965796 --3.411 -3.607372546392 --3.41 -3.60637312858 --3.409 -3.605373711852 --3.408 -3.604374296721 --3.407 -3.603374882683 --3.4059999999999997 -3.602375469994 --3.4050000000000002 -3.60137605891 --3.404 -3.600376648928 --3.403 -3.599377240556 --3.402 -3.598377833295 --3.401 -3.597378427398 --3.4 -3.596379023118 --3.399 -3.595379619961 --3.3979999999999997 -3.594380218178 --3.3970000000000002 -3.59338081802 --3.396 -3.592381418994 --3.395 -3.591382021599 --3.394 -3.590382625344 --3.393 -3.589383230479 --3.392 -3.588383837252 --3.391 -3.587384445177 --3.3899999999999997 -3.586385054502 --3.3890000000000002 -3.585385665473 --3.388 -3.584386277606 --3.387 -3.58338689139 --3.386 -3.582387506346 --3.385 -3.581388122717 --3.384 -3.580388740748 --3.383 -3.579389359961 --3.3819999999999997 -3.578389980837 --3.3810000000000002 -3.577390602905 --3.38 -3.576391226404 --3.379 -3.575391851576 --3.378 -3.57439247795 --3.377 -3.573393105765 --3.376 -3.572393735261 --3.375 -3.571394365971 --3.3739999999999997 -3.570394998366 --3.373 -3.569395631984 --3.372 -3.56839626706 --3.371 -3.56739690383 --3.37 -3.566397541833 --3.3689999999999998 -3.565398181305 --3.368 -3.56439882248 --3.367 -3.5633994649 --3.3659999999999997 -3.562400109028 --3.365 -3.561400754409 --3.364 -3.560401401275 --3.363 -3.559402049859 --3.362 -3.558402699707 --3.3609999999999998 -3.557403351278 --3.36 -3.556404004122 --3.359 -3.555404658468 --3.3579999999999997 -3.554405314546 --3.357 -3.553405971909 --3.356 -3.552406630784 --3.355 -3.551407291401 --3.354 -3.550407953313 --3.3529999999999998 -3.549408616972 --3.352 -3.548409281935 --3.351 -3.54740994843 --3.3499999999999996 -3.546410616679 --3.349 -3.545411286245 --3.348 -3.544411957572 --3.347 -3.543412630224 --3.346 -3.542413304425 --3.3449999999999998 -3.541413980396 --3.344 -3.540414657703 --3.343 -3.53941533657 --3.3419999999999996 -3.538416017217 --3.341 -3.537416699213 --3.34 -3.536417382994 --3.339 -3.535418068133 --3.338 -3.534418754849 --3.3369999999999997 -3.53341944336 --3.336 -3.53242013324 --3.335 -3.531420824709 --3.334 -3.530421517983 --3.333 -3.529422212638 --3.332 -3.528422909105 --3.331 -3.527423606961 --3.33 -3.526424306425 --3.3289999999999997 -3.525425007708 --3.3280000000000003 -3.524425710395 --3.327 -3.523426414909 --3.326 -3.522427120833 --3.325 -3.521427828383 --3.324 -3.520428537769 --3.323 -3.51942924858 --3.322 -3.518429961026 --3.3209999999999997 -3.517430675319 --3.3200000000000003 -3.516431391049 --3.319 -3.515432108632 --3.318 -3.51443282766 --3.317 -3.513433548343 --3.316 -3.5124342708882 --3.315 -3.5114349948921 --3.314 -3.5104357205623 --3.3129999999999997 -3.5094364481054 --3.3120000000000003 -3.5084371771201 --3.311 -3.507437907913 --3.31 -3.5064386402867 --3.309 -3.5054393745472 --3.308 -3.5044401103956 --3.307 -3.5034408478372 --3.306 -3.5024415870761 --3.3049999999999997 -3.5014423280152 --3.3040000000000003 -3.5004430706598 --3.303 -3.499443814913 --3.302 -3.4984445609785 --3.301 -3.4974453087604 --3.3 -3.4964460582622 --3.299 -3.4954468093903 --3.298 -3.4944475623454 --3.2969999999999997 -3.4934483170323 --3.2960000000000003 -3.4924490734562 --3.295 -3.4914498315221 --3.294 -3.4904505914303 --3.293 -3.4894513530866 --3.292 -3.4884521164948 --3.291 -3.487452881563 --3.29 -3.4864536484883 --3.2889999999999997 -3.4854544171774 --3.2880000000000003 -3.4844551876359 --3.287 -3.4834559598659 --3.286 -3.4824567337775 --3.285 -3.481457509565 --3.284 -3.4804582871371 --3.283 -3.4794590664981 --3.282 -3.4784598475577 --3.2809999999999997 -3.4774606305086 --3.2800000000000002 -3.4764614152613 --3.279 -3.4754622018185 --3.278 -3.4744629900919 --3.277 -3.4734637802726 --3.276 -3.4724645722708 --3.275 -3.4714653660911 --3.274 -3.4704661616449 --3.2729999999999997 -3.4694669591214 --3.2720000000000002 -3.4684677584333 --3.271 -3.467468559583 --3.27 -3.4664693624841 --3.269 -3.4654701673244 --3.268 -3.4644709740158 --3.267 -3.4634717825632 --3.266 -3.4624725929704 --3.2649999999999997 -3.4614734051506 --3.2640000000000002 -3.460474219291 --3.263 -3.4594750353036 --3.262 -3.4584758531933 --3.261 -3.4574766728744 --3.26 -3.4564774945312 --3.259 -3.4554783180782 --3.258 -3.4544791435195 --3.2569999999999997 -3.4534799707699 --3.2560000000000002 -3.4524808000135 --3.255 -3.4514816311639 --3.254 -3.4504824642263 --3.253 -3.4494832991165 --3.252 -3.4484841360155 --3.251 -3.4474849748399 --3.25 -3.4464858155937 --3.2489999999999997 -3.4454866582809 --3.248 -3.4444875028198 --3.247 -3.4434883493882 --3.246 -3.4424891979042 --3.245 -3.4414900483717 --3.2439999999999998 -3.4404909007085 --3.243 -3.4394917550931 --3.242 -3.4384926114427 --3.2409999999999997 -3.4374934697616 --3.24 -3.4364943299696 --3.239 -3.4354951922415 --3.238 -3.4344960564971 --3.237 -3.4334969227403 --3.2359999999999998 -3.4324977908906 --3.235 -3.4314986611233 --3.234 -3.4304995333576 --3.2329999999999997 -3.4295004075974 --3.232 -3.4285012837647 --3.231 -3.4275021620307 --3.23 -3.4265030423172 --3.229 -3.4255039246281 --3.2279999999999998 -3.4245048089679 --3.227 -3.4235056952587 --3.226 -3.4225065836715 --3.2249999999999996 -3.4215074741271 --3.224 -3.4205083666314 --3.223 -3.419509261105 --3.222 -3.4185101577191 --3.221 -3.4175110563951 --3.2199999999999998 -3.4165119571375 --3.219 -3.4155128598701 --3.218 -3.4145137647609 --3.2169999999999996 -3.4135146717323 --3.216 -3.4125155807903 --3.215 -3.4115164918571 --3.214 -3.4105174051009 --3.213 -3.4095183204448 --3.2119999999999997 -3.4085192378934 --3.211 -3.407520157372 --3.21 -3.4065210790456 --3.2089999999999996 -3.4055220028386 --3.208 -3.4045229287566 --3.207 -3.4035238568031 --3.206 -3.4025247869048 --3.205 -3.4015257192252 --3.2039999999999997 -3.4005266536891 --3.2030000000000003 -3.3995275903027 --3.202 -3.3985285289907 --3.201 -3.3975294699166 --3.2 -3.3965304130062 --3.199 -3.3955313582642 --3.198 -3.3945323056183 --3.197 -3.393533255229 --3.1959999999999997 -3.3925342070232 --3.1950000000000003 -3.3915351610065 --3.194 -3.3905361171057 --3.193 -3.3895370754813 --3.192 -3.3885380360606 --3.191 -3.3875389988484 --3.19 -3.3865399637741 --3.189 -3.3855409309951 --3.1879999999999997 -3.38454190044 --3.1870000000000003 -3.3835428721146 --3.186 -3.3825438460228 --3.185 -3.3815448220952 --3.184 -3.3805458004879 --3.183 -3.3795467811298 --3.182 -3.3785477640269 --3.181 -3.3775487491091 --3.1799999999999997 -3.3765497365312 --3.1790000000000003 -3.3755507262244 --3.178 -3.3745517181924 --3.177 -3.3735527123674 --3.176 -3.3725537089029 --3.175 -3.3715547077292 --3.174 -3.3705557088525 --3.173 -3.369556712204 --3.1719999999999997 -3.368557717936 --3.1710000000000003 -3.3675587259809 --3.17 -3.3665597363429 --3.169 -3.3655607490282 --3.168 -3.3645617639692 --3.167 -3.3635627813163 --3.166 -3.3625638010027 --3.165 -3.3615648230336 --3.1639999999999997 -3.3605658473419 --3.1630000000000003 -3.3595668740781 --3.162 -3.3585679031741 --3.161 -3.3575689346365 --3.16 -3.3565699683989 --3.159 -3.3555710046093 --3.158 -3.3545720432024 --3.157 -3.3535730841832 --3.1559999999999997 -3.3525741274861 --3.1550000000000002 -3.3515751732592 --3.154 -3.3505762214359 --3.153 -3.3495772720225 --3.152 -3.3485783249544 --3.151 -3.347579380377 --3.15 -3.3465804382262 --3.149 -3.3455814985073 --3.1479999999999997 -3.3445825612255 --3.1470000000000002 -3.3435836263182 --3.146 -3.3425846939283 --3.145 -3.3415857639929 --3.144 -3.3405868365175 --3.143 -3.3395879114387 --3.142 -3.3385889889002 --3.141 -3.3375900688384 --3.1399999999999997 -3.336591151259 --3.1390000000000002 -3.3355922361004 --3.138 -3.3345933235032 --3.137 -3.333594413406 --3.136 -3.3325955058141 --3.135 -3.3315966006657 --3.134 -3.330597698102 --3.133 -3.3295987980609 --3.1319999999999997 -3.3285999005481 --3.1310000000000002 -3.3276010055032 --3.13 -3.3266021130647 --3.129 -3.3256032231723 --3.128 -3.3246043358316 --3.127 -3.3236054510483 --3.126 -3.3226065687625 --3.125 -3.3216076891122 --3.1239999999999997 -3.3206088120368 --3.123 -3.3196099375431 --3.122 -3.3186110655703 --3.121 -3.3176121962564 --3.12 -3.3166133295412 --3.1189999999999998 -3.3156144654306 --3.118 -3.3146156038662 --3.117 -3.3136167449837 --3.1159999999999997 -3.3126178887236 --3.115 -3.3116190350927 --3.114 -3.3106201840321 --3.113 -3.3096213356768 --3.112 -3.3086224899684 --3.1109999999999998 -3.3076236469127 --3.11 -3.3066248064528 --3.109 -3.3056259687217 --3.1079999999999997 -3.3046271336616 --3.107 -3.3036283012794 --3.106 -3.3026294715801 --3.105 -3.301630644508 --3.104 -3.3006318201946 --3.1029999999999998 -3.299632998583 --3.102 -3.2986341796798 --3.101 -3.2976353634287 --3.0999999999999996 -3.2966365499604 --3.099 -3.2956377392195 --3.098 -3.2946389312111 --3.097 -3.2936401258806 --3.096 -3.2926413233576 --3.0949999999999998 -3.2916425235863 --3.094 -3.2906437265734 --3.093 -3.289644932264 --3.0919999999999996 -3.2886461407864 --3.091 -3.2876473520866 --3.09 -3.28664856617 --3.089 -3.2856497830434 --3.088 -3.2846510026529 --3.0869999999999997 -3.2836522251254 --3.086 -3.2826534504074 --3.085 -3.2816546785049 --3.0839999999999996 -3.2806559093644 --3.083 -3.2796571431131 --3.082 -3.2786583796961 --3.081 -3.277659619121 --3.08 -3.2766608613346 --3.0789999999999997 -3.2756621064619 --3.0780000000000003 -3.2746633544508 --3.077 -3.2736646053069 --3.076 -3.2726658589786 --3.075 -3.2716671155901 --3.074 -3.2706683750886 --3.073 -3.2696696374815 --3.072 -3.2686709027167 --3.0709999999999997 -3.2676721709173 --3.0700000000000003 -3.2666734420323 --3.069 -3.2656747160676 --3.068 -3.2646759930303 --3.067 -3.2636772728694 --3.066 -3.2626785557068 --3.065 -3.2616798414919 --3.064 -3.2606811302311 --3.0629999999999997 -3.259682421874 --3.0620000000000003 -3.2586837165423 --3.061 -3.2576850141847 --3.06 -3.2566863148088 --3.059 -3.2556876183644 --3.058 -3.2546889249716 --3.057 -3.2536902345809 --3.056 -3.252691547199 --3.0549999999999997 -3.2516928627764 --3.0540000000000003 -3.250694181433 --3.053 -3.2496955031187 --3.052 -3.248696827841 --3.051 -3.2476981555512 --3.05 -3.2466994863671 --3.049 -3.2457008202405 --3.048 -3.2447021571783 --3.0469999999999997 -3.2437034971872 --3.0460000000000003 -3.2427048402201 --3.045 -3.2417061863929 --3.044 -3.2407075356584 --3.043 -3.2397088880236 --3.042 -3.2387102434407 --3.041 -3.2377116020265 --3.04 -3.236712963733 --3.0389999999999997 -3.2357143285674 --3.0380000000000003 -3.2347156964836 --3.037 -3.2337170675957 --3.036 -3.2327184418577 --3.035 -3.2317198192764 --3.034 -3.2307211998055 --3.033 -3.2297225835597 --3.032 -3.2287239704923 --3.0309999999999997 -3.2277253606105 --3.0300000000000002 -3.2267267538694 --3.029 -3.2257281503812 --3.028 -3.224729550101 --3.027 -3.2237309530358 --3.026 -3.2227323591929 --3.025 -3.2217337685276 --3.024 -3.220735181152 --3.0229999999999997 -3.2197365970208 --3.0220000000000002 -3.2187380161419 --3.021 -3.2177394384703 --3.02 -3.2167408641178 --3.019 -3.2157422930397 --3.018 -3.2147437252433 --3.017 -3.2137451606852 --3.016 -3.2127465994754 --3.0149999999999997 -3.21174804157 --3.0140000000000002 -3.210749486977 --3.013 -3.2097509356523 --3.012 -3.208752387706 --3.011 -3.2077538430947 --3.01 -3.2067553018254 --3.009 -3.2057567638813 --3.008 -3.2047582293203 --3.0069999999999997 -3.2037596980995 --3.0060000000000002 -3.2027611702773 --3.005 -3.2017626458107 --3.004 -3.2007641247579 --3.003 -3.1997656070766 --3.002 -3.1987670928242 --3.001 -3.1977685819591 --3. -3.1967700745385 --2.999 -3.195771570521 --2.9979999999999998 -3.1947730699632 --2.997 -3.1937745728245 --2.996 -3.1927760791613 --2.995 -3.1917775889329 --2.9939999999999998 -3.1907791021956 --2.993 -3.1897806189091 --2.992 -3.1887821391298 --2.991 -3.1877836628171 --2.9899999999999998 -3.1867851900272 --2.989 -3.1857867207443 --2.988 -3.1847882549521 --2.987 -3.1837897927068 --2.9859999999999998 -3.1827913339684 --2.985 -3.181792878793 --2.984 -3.180794427141 --2.983 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--2.66 -2.857539145636146 --2.659 -2.8565424037893137 --2.658 -2.855545669391049 --2.657 -2.85454894245822 --2.656 -2.8535522230077013 --2.655 -2.8525555110564174 --2.654 -2.851558806621535 --2.653 -2.850562109719962 --2.652 -2.849565420368854 --2.651 -2.8485687385851275 --2.65 -2.847572064386224 --2.649 -2.846575397789149 --2.648 -2.8455787388111897 --2.647 -2.84458208746951 --2.646 -2.84358544378156 --2.645 -2.8425888077646326 --2.644 -2.841592179436143 --2.643 -2.8405955588133365 --2.642 -2.839598945913876 --2.641 -2.8386023407606396 --2.64 -2.837605743360288 --2.639 -2.83660915373575 --2.638 -2.8356125719047647 --2.637 -2.834615997884926 --2.636 -2.8336194316940144 --2.635 -2.8326228733495302 --2.634 -2.831626322869512 --2.633 -2.8306297802716753 --2.632 -2.829633245573833 --2.631 -2.828636718793841 --2.63 -2.8276401999497733 --2.629 -2.826643689059517 --2.628 -2.8256471861410843 --2.627 -2.8246506912124656 --2.626 -2.823654204291846 --2.625 -2.8226577253974385 --2.624 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--2.5909999999999997 -2.788782350686307 --2.59 -2.7877861644473776 --2.589 -2.7867899869148567 --2.588 -2.7857938181083526 --2.5869999999999997 -2.7847976580475526 --2.586 -2.7838015067524142 --2.585 -2.782805364242612 --2.584 -2.781809230538028 --2.5829999999999997 -2.780813105658482 --2.582 -2.7798169896241145 --2.581 -2.778820882459468 --2.58 -2.777824784175215 --2.5789999999999997 -2.7768286947960203 --2.578 -2.775832614342236 --2.577 -2.7748365428338766 --2.576 -2.7738404802912133 --2.5749999999999997 -2.772844426734379 --2.574 -2.771848382183839 --2.573 -2.7708523466599146 --2.572 -2.7698563201829094 --2.5709999999999997 -2.7688603027732617 --2.57 -2.7678642944515586 --2.569 -2.766868295238226 --2.568 -2.7658723051538905 --2.567 -2.764876324219167 --2.566 -2.7638803524546143 --2.565 -2.762884389881115 --2.564 -2.7618884365193326 --2.563 -2.7608924923900626 --2.562 -2.759896557514118 --2.561 -2.7589006319169687 --2.56 -2.7579047156105028 --2.559 -2.7569088086202687 --2.558 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--1.9779999999999998 -2.1808394954993995 --1.9769999999999999 -2.1798546885690624 --1.976 -2.1788699153376765 --1.975 -2.1778851758771354 --1.9739999999999998 -2.176900470259476 --1.9729999999999999 -2.175915798556842 --1.972 -2.174931160841542 --1.971 -2.1739465571860372 --1.9699999999999998 -2.1729619876629016 --1.9689999999999999 -2.1719774523448745 --1.968 -2.1709929513048523 --1.967 -2.1700084846158028 --1.9659999999999997 -2.1690240523508963 --1.9649999999999999 -2.1680396545834864 --1.964 -2.1670552913869643 --1.963 -2.166070962834937 --1.9619999999999997 -2.165086669001135 --1.9609999999999999 -2.1641024099594555 --1.96 -2.163118185783866 --1.959 -2.1621339965485955 --1.9579999999999997 -2.1611498423279207 --1.9569999999999999 -2.160165723196297 --1.956 -2.1591816392283163 --1.955 -2.158197590498745 --1.9539999999999997 -2.157213577082446 --1.9529999999999998 -2.156229599054493 --1.952 -2.1552456564900027 --1.951 -2.154261749464383 --1.9499999999999997 -2.1532778780530646 --1.9489999999999998 -2.1522940423316843 --1.948 -2.1513102423759753 --1.947 -2.1503264782618627 --1.9459999999999997 -2.1493427500654514 --1.9449999999999998 -2.148359057862903 --1.944 -2.1473754017305837 --1.943 -2.1463917817450193 --1.9419999999999997 -2.1454081979828628 --1.9409999999999998 -2.144424650520893 --1.94 -2.1434411394360597 --1.939 -2.1424576648054887 --1.9379999999999997 -2.1414742267063813 --1.9369999999999998 -2.1404908252161814 --1.936 -2.1395074604124424 --1.935 -2.138524132372827 --1.9339999999999997 -2.1375408411751753 --1.9329999999999998 -2.1365575868975233 --1.932 -2.1355743696179865 --1.931 -2.134591189414893 --1.9299999999999997 -2.133608046366639 --1.9289999999999998 -2.1326249405518958 --1.928 -2.131641872049356 --1.927 -2.130658840937957 --1.9260000000000002 -2.129675847296738 --1.9249999999999998 -2.128692891204903 --1.924 -2.1277099727418305 --1.923 -2.1267270919869996 --1.9220000000000002 -2.1257442490201095 --1.9209999999999998 -2.124761443920974 --1.92 -2.1237786767695392 --1.919 -2.1227959476459493 --1.9180000000000001 -2.1218132566304666 --1.9169999999999998 -2.1208306038035527 --1.916 -2.119847989245758 --1.915 -2.1188654130378373 --1.9140000000000001 -2.1178828752606993 --1.9129999999999998 -2.1169003759953706 --1.912 -2.1159179153230743 --1.911 -2.1149354933251576 --1.9100000000000001 -2.1139531100831364 --1.9089999999999998 -2.1129707656786905 --1.908 -2.111988460193632 --1.907 -2.1110061937099647 --1.9060000000000001 -2.1100239663098193 --1.9049999999999998 -2.1090417780754693 --1.904 -2.1080596290894085 --1.903 -2.1070775194342235 --1.9020000000000001 -2.1060954491926895 --1.9009999999999998 -2.1051134184477274 --1.9 -2.1041314272824256 --1.899 -2.1031494757800067 --1.8980000000000001 -2.1021675640238864 --1.8969999999999998 -2.1011856920976264 --1.896 -2.100203860084932 --1.895 -2.0992220680696696 --1.8940000000000001 -2.0982403161359207 --1.8929999999999998 -2.097258604367833 --1.892 -2.0962769328497672 --1.891 -2.0952953016662352 --1.8900000000000001 -2.094313710901913 --1.8889999999999998 -2.0933321606416486 --1.888 -2.0923506509704297 --1.887 -2.091369181973396 --1.8860000000000001 -2.0903877537358815 --1.8849999999999998 -2.0894063663433657 --1.884 -2.088425019881479 --1.883 -2.087443714436008 --1.8820000000000001 -2.086462450092942 --1.8809999999999998 -2.0854812269383842 --1.88 -2.08450004505863 --1.879 -2.0835189045401385 --1.8780000000000001 -2.0825378054695203 --1.8769999999999998 -2.081556747933533 --1.876 -2.0805757320191285 --1.875 -2.079594757813409 --1.874 -2.078613825403638 --1.8729999999999998 -2.0776329348772538 --1.8719999999999999 -2.0766520863218294 --1.871 -2.0756712798251677 --1.87 -2.0746905154751456 --1.8689999999999998 -2.0737097933598645 --1.8679999999999999 -2.072729113567589 --1.867 -2.0717484761867353 --1.866 -2.0707678813059025 --1.8649999999999998 -2.069787329013789 --1.8639999999999999 -2.0688068193993745 --1.863 -2.067826352551699 --1.862 -2.06684592856003 --1.8609999999999998 -2.0658655475137953 --1.8599999999999999 -2.0648852095025494 --1.859 -2.0639049146160793 --1.858 -2.0629246629442717 --1.8569999999999998 -2.061944454577221 --1.8559999999999999 -2.0609642896051787 --1.855 -2.059984168118574 --1.854 -2.059004090207995 --1.8529999999999998 -2.058024055964203 --1.8519999999999999 -2.0570440654781144 --1.851 -2.0560641188408315 --1.85 -2.0550842161436353 --1.8489999999999998 -2.0541043574779523 --1.8479999999999999 -2.053124542935375 --1.847 -2.0521447726076913 --1.846 -2.051165046586841 --1.8449999999999998 -2.050185364964948 --1.8439999999999999 -2.049205727834289 --1.843 -2.0482261352873254 --1.842 -2.0472465874167076 --1.8409999999999997 -2.0462670843152013 --1.8399999999999999 -2.045287626075789 --1.839 -2.0443082127916274 --1.838 -2.043328844556019 --1.8369999999999997 -2.0423495214624663 --1.8359999999999999 -2.041370243604611 --1.835 -2.0403910110762777 --1.834 -2.0394118239715184 --1.8329999999999997 -2.0384326823844745 --1.8319999999999999 -2.0374535864095087 --1.831 -2.0364745361411485 --1.83 -2.0354955316740897 --1.8289999999999997 -2.034516573103205 --1.8279999999999998 -2.0335376605235678 --1.827 -2.032558794030379 --1.826 -2.0315799737190408 --1.8249999999999997 -2.0306011996851105 --1.8239999999999998 -2.029622472024365 --1.823 -2.0286437908327155 --1.822 -2.0276651562062638 --1.8209999999999997 -2.026686568241287 --1.8199999999999998 -2.0257080270342342 --1.819 -2.024729532681726 --1.818 -2.023751085280584 --1.8169999999999997 -2.0227726849277743 --1.8159999999999998 -2.0217943317204705 --1.815 -2.0208160257559884 --1.814 -2.019837767131869 --1.8129999999999997 -2.018859555945788 --1.8119999999999998 -2.0178813922956094 --1.811 -2.0169032762794252 --1.81 -2.0159252079953767 --1.8089999999999997 -2.0149471875419556 --1.8079999999999998 -2.013969215017682 --1.807 -2.0129912905213465 --1.806 -2.012013414151892 --1.8049999999999997 -2.011035586008439 --1.8039999999999998 -2.0100578061902863 --1.803 -2.009080074796922 --1.802 -2.008102391928005 --1.8009999999999997 -2.007124757683371 --1.7999999999999998 -2.0061471721630317 --1.799 -2.005169635467232 --1.798 -2.004192147696341 --1.7970000000000002 -2.003214708950901 --1.7959999999999998 -2.0022373193316816 --1.795 -2.001259978939598 --1.794 -2.000282687875776 --1.7930000000000001 -1.9993054462415178 --1.7919999999999998 -1.9983282541382779 --1.791 -1.9973511116677354 --1.79 -1.9963740189317192 --1.7890000000000001 -1.9953969760322618 --1.7879999999999998 -1.9944199830715819 --1.787 -1.9934430401520422 --1.786 -1.9924661473762566 --1.7850000000000001 -1.9914893048469509 --1.7839999999999998 -1.9905125126671037 --1.783 -1.9895357709398156 --1.782 -1.9885590797684232 --1.7810000000000001 -1.9875824392564008 --1.7799999999999998 -1.9866058495074645 --1.779 -1.9856293106254574 --1.778 -1.9846528227144502 --1.7770000000000001 -1.9836763858786872 --1.7759999999999998 -1.9827000002225768 --1.775 -1.9817236658507595 --1.774 -1.9807473828679993 --1.7730000000000001 -1.9797711513793161 --1.7719999999999998 -1.9787949714898612 --1.771 -1.977818843305014 --1.77 -1.9768427669303186 --1.7690000000000001 -1.9758667424714966 --1.7679999999999998 -1.974890770034487 --1.767 -1.9739148497253995 --1.766 -1.972938981650537 --1.7650000000000001 -1.9719631659163694 --1.7639999999999998 -1.9709874026295935 --1.763 -1.9700116918970707 --1.762 -1.9690360338258608 --1.7610000000000001 -1.9680604285232048 --1.7599999999999998 -1.9670848760965305 --1.759 -1.966109376653478 --1.758 -1.9651339303018394 --1.7570000000000001 -1.9641585371496342 --1.7559999999999998 -1.963183197305043 --1.755 -1.9622079108764818 --1.754 -1.9612326779724945 --1.7530000000000001 -1.9602574987018584 --1.7519999999999998 -1.959282373173549 --1.751 -1.9583073014967018 --1.75 -1.9573322837806444 --1.749 -1.9563573201349522 --1.7479999999999998 -1.9553824106693052 --1.7469999999999999 -1.9544075554936546 --1.746 -1.9534327547180916 --1.745 -1.9524580084529384 --1.7439999999999998 -1.9514833168086672 --1.7429999999999999 -1.9505086798959936 --1.742 -1.949534097825804 --1.741 -1.9485595707091523 --1.7399999999999998 -1.9475850986572962 --1.7389999999999999 -1.9466106817817537 --1.738 -1.9456363201941576 --1.737 -1.9446620140063569 --1.7359999999999998 -1.9436877633304073 --1.7349999999999999 -1.9427135682785326 --1.734 -1.941739428963212 --1.733 -1.940765345497041 --1.7319999999999998 -1.9397913179928703 --1.7309999999999999 -1.938817346563715 --1.73 -1.9378434313228057 --1.729 -1.9368695723835718 --1.7279999999999998 -1.9358957698595889 --1.7269999999999999 -1.93492202386471 --1.726 -1.9339483345129191 --1.725 -1.9329747019184338 --1.7239999999999998 -1.9320011261956438 --1.7229999999999999 -1.9310276074591601 --1.722 -1.9300541458237623 --1.721 -1.9290807414044715 --1.7199999999999998 -1.9281073943164537 --1.7189999999999999 -1.9271341046751134 --1.718 -1.926160872596045 --1.717 -1.9251876981950273 --1.7159999999999997 -1.9242145815880531 --1.7149999999999999 -1.9232415228913102 --1.714 -1.9222685222211906 --1.713 -1.9212955796942535 --1.7119999999999997 -1.9203226954273145 --1.7109999999999999 -1.9193498695373337 --1.71 -1.9183771021415221 --1.709 -1.9174043933572413 --1.7079999999999997 -1.9164317433020879 --1.7069999999999999 -1.9154591520938529 --1.706 -1.9144866198505124 --1.705 -1.9135141466902577 --1.7039999999999997 -1.9125417327314833 --1.7029999999999998 -1.9115693780927718 --1.702 -1.9105970828929255 --1.701 -1.9096248472509503 --1.6999999999999997 -1.9086526712860086 --1.6989999999999998 -1.9076805551175322 --1.698 -1.9067084988651086 --1.697 -1.9057365026485453 --1.6959999999999997 -1.90476456658785 --1.6949999999999998 -1.9037926908032192 --1.694 -1.9028208754151021 --1.693 -1.9018491205440717 --1.6919999999999997 -1.9008774263109904 --1.6909999999999998 -1.8999057928368561 --1.69 -1.8989342202429027 --1.689 -1.897962708650572 --1.6879999999999997 -1.8969912581814998 --1.6869999999999998 -1.89601986895751 --1.686 -1.895048541100674 --1.685 -1.8940772747332233 --1.6839999999999997 -1.8931060699776365 --1.6829999999999998 -1.8921349269565724 --1.682 -1.8911638457928814 --1.681 -1.8901928266096526 --1.6799999999999997 -1.8892218695301604 --1.6789999999999998 -1.8882509746779013 --1.678 -1.8872801421765564 --1.677 -1.8863093721500277 --1.6759999999999997 -1.8853386647224264 --1.6749999999999998 -1.8843680200180613 --1.674 -1.883397438161451 --1.673 -1.8824269192773238 --1.6719999999999997 -1.881456463490623 --1.6709999999999998 -1.8804860709264828 --1.67 -1.8795157417102482 --1.669 -1.8785454759674836 --1.6680000000000001 -1.8775752738239624 --1.6669999999999998 -1.8766051354056534 --1.666 -1.8756350608387367 --1.665 -1.874665050249613 --1.6640000000000001 -1.873695103764886 --1.6629999999999998 -1.8727252215113477 --1.662 -1.8717554036160367 --1.661 -1.8707856502061782 --1.6600000000000001 -1.8698159614092122 --1.6589999999999998 -1.8688463373527706 --1.658 -1.867876778164736 --1.657 -1.8669072839731728 --1.6560000000000001 -1.8659378549063534 --1.6549999999999998 -1.8649684910927617 --1.654 -1.863999192661121 --1.653 -1.8630299597403222 --1.6520000000000001 -1.862060792459494 --1.6509999999999998 -1.8610916909479736 --1.65 -1.8601226553353145 --1.649 -1.8591536857512532 --1.6480000000000001 -1.8581847823257909 --1.6469999999999998 -1.8572159451890706 --1.646 -1.8562471744715103 --1.645 -1.8552784703037069 --1.6440000000000001 -1.854309832816474 --1.6429999999999998 -1.853341262140867 --1.642 -1.8523727584080998 --1.641 -1.8514043217496479 --1.6400000000000001 -1.8504359522971654 --1.6389999999999998 -1.849467650182543 --1.638 -1.848499415537878 --1.637 -1.8475312484954847 --1.6360000000000001 -1.8465631491878902 --1.6349999999999998 -1.8455951177478205 --1.634 -1.844627154308229 --1.633 -1.8436592590022944 --1.6320000000000001 -1.842691431963383 --1.6309999999999998 -1.841723673325088 --1.63 -1.8407559832212352 --1.629 -1.8397883617858604 --1.6280000000000001 -1.8388208091531653 --1.6269999999999998 -1.837853325457645 --1.626 -1.8368859108339415 --1.625 -1.8359185654169574 --1.624 -1.8349512893417956 --1.6229999999999998 -1.8339840827437683 --1.6219999999999999 -1.833016945758414 --1.621 -1.8320498785214883 --1.62 -1.831082881168949 --1.6189999999999998 -1.8301159538369869 --1.6179999999999999 -1.829149096661997 --1.617 -1.8281823097806014 --1.616 -1.827215593329637 --1.6149999999999998 -1.8262489474461558 --1.6139999999999999 -1.825282372267422 --1.613 -1.824315867930919 --1.612 -1.8233494345743553 --1.6109999999999998 -1.8223830723356562 --1.6099999999999999 -1.8214167813529643 --1.609 -1.8204505617646327 --1.608 -1.8194844137092325 --1.6069999999999998 -1.8185183373255587 --1.6059999999999999 -1.8175523327526437 --1.605 -1.8165864001296956 --1.604 -1.8156205395961655 --1.6029999999999998 -1.8146547512917441 --1.6019999999999999 -1.8136890353563073 --1.601 -1.812723391929959 --1.6 -1.8117578211530363 --1.5989999999999998 -1.810792323166073 --1.5979999999999999 -1.8098268981098475 --1.597 -1.8088615461253352 --1.596 -1.8078962673537562 --1.5949999999999998 -1.8069310619365189 --1.5939999999999999 -1.8059659300152986 --1.593 -1.8050008717319197 --1.592 -1.8040358872285052 --1.5909999999999997 -1.8030709766473607 --1.5899999999999999 -1.8021061401309963 --1.589 -1.8011413778221652 --1.588 -1.800176689863853 --1.5869999999999997 -1.7992120763992379 --1.5859999999999999 -1.7982475375717362 --1.585 -1.7972830735249856 --1.584 -1.796318684402843 --1.5829999999999997 -1.795354370349378 --1.5819999999999999 -1.794390131508899 --1.581 -1.7934259680259097 --1.58 -1.792461880045183 --1.5789999999999997 -1.7914978677116726 --1.5779999999999998 -1.79053393117055 --1.577 -1.7895700705672453 --1.576 -1.788606286047387 --1.5749999999999997 -1.787642577756818 --1.5739999999999998 -1.7866789458416332 --1.573 -1.7857153904481278 --1.572 -1.7847519117228239 --1.5709999999999997 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"#8fb032" -cc4 = "#eb6235" - -set logscale y - -data = "figs/fig-series-data.dat" - -set xrange [0.5:15.5] -set yrange [0.000005:5] -set key off -set xlabel '$n$' -set ylabel offset 2 '$|f_n|$' - -set lmargin 7 - -set ytics format '' - -set ytics add ('$\footnotesize10^{-5}$' 10**(-5),'$\footnotesize10^{-4}$' 10**(-4), '$\footnotesize10^{-3}$' 10**(-3),'$\footnotesize10^{-2}$' 10**(-2), '$\footnotesize10^{-1}$' 10**(-1),'$\footnotesize10^{0}$' 10**(0)) - -plot \ - data using 1:3 with points lc rgb cc1, \ - data using 1:2 with points lc rgb cc2, \ - data using 1:5 with points lc rgb cc3 - diff --git a/figs/fig-sus_scaling-func-mod.dat b/figs/fig-sus_scaling-func-mod.dat deleted file mode 100644 index 5d49f9b..0000000 --- a/figs/fig-sus_scaling-func-mod.dat +++ /dev/null @@ -1,11515 +0,0 @@ -0.0004967499652960352 0.02552916136980551 -0.0009137823733494827 0.02552090801819801 -0.0013310520041847056 0.025513600697881153 -0.0017485589391690029 0.025506236977519796 -0.0021663032596971267 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terminal pslatex rotate size 3.417,4.222 -set multiplot - -cc1 = "#5e81b5" -cc2 = "#e19c24" -cc3 = "#8fb032" -cc4 = "#eb6235" - -Tc = 2 / log(1 + sqrt(2)) -Delta = 15. / 8 -gamma = 7. / 4 -beta = 1. / 8 - -Ch = 0.838677624411 - -t(T) = abs((T - Tc) / T) -h(T, H) = H / T -X(T, H) = h(T, H) * t(T)**(-Delta) - -poly(A, l, m, X) = A * X**m / (1 + (l * X)**(m + 1)) -polyint(A, l, m, X) = A * l**(-(m + 1)) * log(1 + (l * X)**(m + 1)) / (m + 1) - -G(i) = i == 1 ? -1.3578383417066 : \ - i == 2 ? -0.048953289720 : \ - i == 3 ? 0.038863932 : \ - i == 4 ? -0.068362119 : \ - i == 5 ? 0.18388370 : \ - i == 6 ? -0.6591714 : \ - i == 7 ? 2.937665 : \ - i == 8 ? -15.61 : 0 -GC(i) = G(i) * (2 * asinh(1))**2 * (Ch * (2 * asinh(1))**(-Delta))**i - -M0 = (2**2.5 * asinh(1))**0.125 -B = Tc**2 * M0 / (16 * pi) -C0 = 0.0255369719 -A = pi / 2 * C0 / (B**2) - -n = 1 -c(i) = i == 1 ? 0.00856277 : 0 -lamb = 10.487 -A2 = 1.70034 -B2 = B - -n2 = 5 -c2(i) = i == 1 ? 0.00639522 : \ - i == 2 ? 0.00143461 : \ - i == 3 ? -0.206879 : \ - i == 4 ? 4.39278 : \ - i == 5 ? -56.1389 : 0 -lamb2 = 13.2489 -A3 = 1.91747 -B3 = B - -susfunc = "figs/fig-sus_scaling-func.dat" -susfuncm = "figs/fig-sus_scaling-func-mod.dat" -magfunc = "figs/fig-mag_scaling-func.dat" -magfuncm = "figs/fig-mag_scaling-func-mod.dat" - -num = "data/data_square-4096.dat" - -set samples 20000 -set key off - -set size 1,1 - 0.52 -set origin 0,0.52 -set xrange [-40:1200] -set yrange [-1:33] -set ylabel offset 1,0 '$T\chi|t|^{\gamma}\times 10^3$' -set lmargin 6 -set xtics format '' -set mxtics 5 -set mytics 5 -set bmargin 0.2 - -plot \ - num using (X($2, $3)):(10**3 * Tc * $10 * t($2)**gamma):(10**3 * Tc * $11 * t($2)**gamma) with yerrorbars pt 0 lc black, \ - susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 3 lc rgb cc1 dt 1, \ - susfuncm using 1:(10**3 * $2) with lines lw 3 lc rgb cc2 dt 2, \ - -10**3 * (sum[i=1:7] GC(i + 1) * i * (i + 1) * (x)**(i-1)) with lines lw 3 lc rgb cc3 dt 3 - -set bmargin -1 -set tmargin 0.2 -set size 1,0.52 -set origin 0,0 -set yrange [1.18:1.74] -set ylabel offset 1,0 '$M|t|^{-\beta}$' -set xlabel '$h|t|^{-\beta\delta}$' -set xtics format '%g' - -plot \ - num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)) with yerrorbars pt 0 lc black, \ - magfunc using (10**$1 / B):(M0 + 10**($2) * A * B) with linespoints pt 0 lw 3 lc rgb cc1 dt 1, \ - magfuncm using 1:2 with lines lw 3 lc rgb cc2 dt 2, \ - -sum[i=1:8] GC(i) * i * x**(i-1) with lines lw 3 lc rgb cc3 dt 3 - -set logscale xy -set tmargin -1 -set lmargin -1 -set size 0.7,0.325 -set origin 0.26,0.5 + 0.29 / 2 -set xrange [0.0015:1900] -set yrange [0.00002:0.08] -set ylabel offset 4,0 '\footnotesize$T\chi|t|^\gamma$' -set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\beta\delta}$' -set mxtics 5 -#set xtics format '' -2,10,1000 -set xtics add ('$\footnotesize10^{-2}$' 10**(-2), "" 0.1, '$\footnotesize10^0$' 1, "" 10, '$\footnotesize10^2$' 100, "" 1000) -set mytics 5 -#set ytics format '\footnotesize$10^{%T}$' 0.00001,10,0.01 -set ytics add ('$\footnotesize10^{-4}$' 10**(-4), '$\footnotesize10^{-3}$' 10**(-3), '$\footnotesize10^{-2}$' 10**(-2)) - -plot \ - num using (X($2, $3)):(Tc * $10 * t($2)**gamma):(Tc * $11 * t($2)**gamma) with yerrorbars pt 0 lc black, \ - susfunc using (10**$1 / B):(10**$2 * A * B**2) with linespoints pt 0 lw 3 lc rgb cc1 dt 1, \ - susfuncm using 1:2 with lines lw 3 lc rgb cc2 dt 2, \ - -sum[i=2:8] GC(i) * i * (i-1) * x**(i-2) with lines dt 3 lw 3 lc rgb cc3 - -unset logscale xy -set logscale x -set size 0.65,0.325 -set origin 0.31,0.23 / 2 -set yrange [1.18:1.74] -set ylabel offset 4,0 '\footnotesize$M|t|^{-\beta}$' -set ytics format '\footnotesize {%g}' 1.2,0.2,1.8 -set mytics 5 - -plot \ - num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)) with yerrorbars pt 0 lc black, \ - magfunc using (10**$1 / B):(M0 + A * B * 10**$2) with linespoints pt 0 lw 3 lc rgb cc1 dt 1, \ - magfuncm using 1:2 with lines lw 3 lc rgb cc2 dt 2, \ - -sum[i=1:8] GC(i) * i * x**(i-1) with lines lw 3 lc rgb cc3 dt 3 - diff --git a/makefile b/makefile deleted file mode 100644 index a04aaa6..0000000 --- a/makefile +++ /dev/null @@ -1,21 +0,0 @@ - -DOC=essential-ising -FIGS=fig-susmag fig-series -DATA=data_square-4096 -FUNCS=fig-mag_scaling-func fig-sus_scaling-func fig-series-data - -all: ${DOC}.pdf - -%.tex: figs/%.gplot ${DATA:%=data/%.dat} ${FUNCS:%=figs/%.dat} - gnuplot $< > $@ - -${DOC}.pdf: ${DOC}.tex ${DOC}.bib ${FIGS:%=%.tex} - rubber $(DOC).tex - dvipdf $(DOC).dvi - -clean: - rubber --clean $(DOC) - rm -f $(DOC).pdf - rm -f $(DOC)Notes.bib - rm -f ${FIGS:%=%.tex} - diff --git a/refs/PhysRevB.40.9006.pdf b/refs/PhysRevB.40.9006.pdf deleted file mode 100644 index d455145..0000000 Binary files a/refs/PhysRevB.40.9006.pdf and /dev/null differ -- cgit v1.2.3-70-g09d2 From 8c7ba65464178104b3bb68e1a15b1e6fbf3e3231 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 22 Dec 2020 19:45:59 +0100 Subject: Started new paper. --- .gitignore | 4 + essential-ising.tex | 478 ---------------------------------------------------- ising_scaling.bib | 57 +++++++ ising_scaling.tex | 42 +++++ 4 files changed, 103 insertions(+), 478 deletions(-) delete mode 100644 essential-ising.tex create mode 100644 ising_scaling.bib create mode 100644 ising_scaling.tex diff --git a/.gitignore b/.gitignore index 034b355..84b12c2 100644 --- a/.gitignore +++ b/.gitignore @@ -4,4 +4,8 @@ *.blg *Notes.bib *.dvi +*.fdb_latexmk +*.synctex.gz +*.fls +*.out /*.pdf diff --git a/essential-ising.tex b/essential-ising.tex deleted file mode 100644 index e125229..0000000 --- a/essential-ising.tex +++ /dev/null @@ -1,478 +0,0 @@ - -% -% Created by Jaron Kent-Dobias on Thu Apr 20 12:50:56 EDT 2017. -% Copyright (c) 2017 Jaron Kent-Dobias. All rights reserved. -% -\documentclass[aps,prl,reprint]{revtex4-1} - -\usepackage[utf8]{inputenc} -\usepackage{amsmath,amssymb,latexsym,mathtools,xifthen} - -% uncomment to label only equations that are referenced in the text -%\mathtoolsset{showonlyrefs=true} - -% I want labels but don't want to type out ``equation'' -\def\[{\begin{equation}} -\def\]{\end{equation}} - -% math not built-in -\def\arcsinh{\mathop{\mathrm{arcsinh}}\nolimits} -\def\arccosh{\mathop{\mathrm{arccosh}}\nolimits} -\def\ei{\mathop{\mathrm{Ei}}\nolimits} % exponential integral Ei -\def\re{\mathop{\mathrm{Re}}\nolimits} -\def\im{\mathop{\mathrm{Im}}\nolimits} -\def\sgn{\mathop{\mathrm{sgn}}\nolimits} -\def\dd{d} % differential -\def\O{O} % big O -\def\o{o} % little O - -% subscript for ``critical'' values, e.g., T_\c -\def\c{\mathrm c} - -% scaling functions -\def\fM{\mathcal M} % magnetization -\def\fX{\mathcal Y} % susceptibility -\def\fXt{\check{\mathcal Y}} % susceptibility -\def\fF{\mathcal F} % free energy -\def\fiF{\mathcal H} % imaginary free energy -\def\fS{\mathcal S} % surface tension -\def\fG{\mathcal G} % exponential factor - -% lattice types -\def\sq{\mathrm{sq}} -\def\tri{\mathrm{tri}} -\def\hex{\mathrm{hex}} - -% dimensions -\def\dim{d} -\def\twodee{\textsc{2d}} -\def\threedee{\textsc{3d}} -\def\fourdee{\textsc{4d}} - -% fancy partial derivative -\newcommand\pd[3][]{ - \ifthenelse{\isempty{#1}} - {\def\tmp{}} - {\def\tmp{^#1}} - \frac{\partial\tmp#2}{\partial#3\tmp} -} - -% used to reformat display math to fit in two-column ``reprint'' mode -\makeatletter -\newif\ifreprint -\@ifclasswith{revtex4-1}{reprint}{\reprinttrue}{\reprintfalse} -\makeatother - -\begin{document} - -\title{Essential Singularities in Universal Scaling Functions at the Ising Coexistence Line} -\author{Jaron Kent-Dobias} -\author{James P.~Sethna} -\affiliation{Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA} - -\date\today - -\begin{abstract} - Renormalization group ideas and results from critical droplet theory are - used to construct a scaling ansatz for the imaginary component of the free - energy of an Ising model in its metastable state close to the critical - point. The analytic properties of the free energy are used to determine - scaling functions for the free energy in the vicinity of the critical point - and the abrupt transition line. These functions have essential singularities - at the abrupt transition. Analogous forms for the magnetization and susceptibility in - two dimensions are fit to numeric data and show good agreement, especially - when their nonsingular behavior is modified to match existing numeric results. -\end{abstract} - -\maketitle - -The Ising model is the canonical example of a system with a continuous phase -transition, and the study of its singular properties marked the first success -of the renormalization group (\textsc{rg}) method in statistical physics -\cite{wilson.1971.renormalization}. Its status makes sense: it's a simple -model whose continuous phase transition contains all the essential features of -more complex ones, but admits \textsc{rg} methods in a straightforward way and -has exact solutions in certain dimensions and for certain parameter -restrictions. However, the Ising critical point is not simply a continuous -transition: it also ends a line of abrupt phase transitions extending from it -at zero field below the critical temperature. Though typically neglected in -\textsc{rg} scaling analyses of the critical point, we demonstrate that there -are numerically measurable contributions to scaling due to the abrupt -transition line that cannot be accounted for by analytic changes of control or -thermodynamic variables. - -\textsc{Rg} analysis predicts that the singular part of the free energy per -site $F$ as a function of reduced temperature $t=1-T_\c/T$ and field $h=H/T$ -in the vicinity of the critical point takes the scaling form -$F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})$ for the low temperature phase -$t<0$ \cite{cardy.1996.scaling}. When studying the properties of the Ising -critical point, it is nearly always assumed that the universal scaling -function $\fF$ is analytic, i.e., has a convergent Taylor series. However, it -has long been known that there exists an essential singularity in $\fF$ at -zero argument, though its effects have long been believed to be unobservable -\cite{fisher.1967.condensation}, or simply just neglected -\cite{guida.1997.3dising,schofield.1969.parametric,schofield.1969.correlation,caselle.2001.critical,josephson.1969.equation,fisher.1999.trigonometric}. -With careful analysis, we have found that assuming the presence of the -essential singularity is predictive of the scaling form of, for instance, the -susceptibility and magnetization. - -The provenance of the essential singularity can be understood using the -methods of critical droplet theory for the decay of an Ising system in a -metastable state, i.e., an equilibrium Ising state for $T0$ -subjected to a small negative external field $H<0$. The existence of an -essential singularity has also been suggested by transfer matrix -\cite{mccraw.1978.metastability,enting.1980.investigation,mangazeev.2008.variational,mangazeev.2010.scaling} and \textsc{rg} -methods \cite{klein.1976.essential}, and a different kind of essential -singularity is known to exist in the zero-temperature susceptibility -\cite{orrick.2001.susceptibility,chan.2011.ising,guttmann.1996.solvability,nickel.1999.singularity,nickel.2000.addendum,assis.2017.analyticity}. It has long been known that the decay -rate $\Gamma$ of metastable states in statistical mechanics is often related -to the metastable free energy $F$ by $\Gamma\propto\im F$ -\cite{langer.1969.metastable,penrose.1987.rigorous,gaveau.1989.analytic,privman.1982.analytic}. -`Metastable free energy' can be thought of as either an analytic continuation -of the free energy through the abrupt phase transition, or restriction of the -partition function trace to states in the vicinity of the local free energy -minimum that characterizes the metastable state. In any case, the free energy -develops a nonzero imaginary part in the metastable region. Heuristically, -this can be thought of as similar to what happens in quantum mechanics with a -non-unitary Hamiltonian: the imaginary part describes loss rate of probability -that the system occupies any `accessible' state, which corresponds to decay. - -In critical droplet theory, the metastable state decays when a domain of the -equilibrium state forms whose surface-energy cost for growth is outweighed by -bulk-energy gains. There is numerical evidence that, near the critical point, -these droplets are spherical \cite{gunther.1993.transfer-matrix}. The free energy -cost of the surface of a droplet of radius $R$ is $\Sigma S_\dim R^{\dim-1}$ -and that of its bulk is $-M|H|V_\dim R^\dim$, where $S_\dim$ and $V_\dim$ are -the surface area and volume of a $(\dim-1)$-sphere, respectively, and $\Sigma$ -is the surface tension of the equilibrium--metastable interface. The critical -droplet size is then $R_\c=(\dim-1)\Sigma/M|H|$ and the free energy of the -critical droplet is $\Delta F_\c=\pi^{\dim/2}\Sigma^\dim((\dim-1)/M|H|)^{\dim-1}/\Gamma(1+\dim/2)$. -Assuming the typical singular scaling forms -$\Sigma/T=|t|^\mu\fS(h|t|^{-\beta\delta})$ and $M=|t|^\beta\mathcal -M(h|t|^{-\beta\delta})$ and using known hyperscaling relations -\cite{widom.1981.interface}, this implies a scaling form -\def\eqcritformone{ - T\frac{\pi^{\dim/2}(\dim-1)^{\dim-1}}{\Gamma(1+\dim/2)} - \frac{\fS^\dim(h|t|^{-\beta\delta})}{(-h|t|^{-\beta\delta} - \fM(h|t|^{-\beta\delta}))^{\dim-1}} -} -\def\eqcritformtwo{ - T\fG^{-(\dim-1)}(h|t|^{-\beta\delta}) -} -\ifreprint -\[ - \begin{aligned} - \Delta F_\c - &=\eqcritformone\\ - &\sim\eqcritformtwo - \end{aligned} -\] -\else -\[ - \Delta F_\c=\eqcritformone\sim\eqcritformtwo -\] -\fi -for the free energy change due to the critical droplet. -Since both surface tension and magnetization are finite and nonzero for $H=0$ -at $T1$ this function has an essential singularity in the invariant -combination $h|t|^{-\beta\delta}$. - -This form of $\im F$ for small $h$ is well known -\cite{langer.1967.condensation,harris.1984.metastability}. We make the -scaling ansatz that the imaginary part of the metastable free energy has the -same singular behavior as the real part of the equilibrium free energy, and -that for small $t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\fiF(h|t|^{-\beta\delta})$, -where -\[ - \fiF(X)=A\Theta(-X)(-BX)^be^{-1/(-BX)^{\dim-1}} - \label{eq:im.scaling} -\] -and $\Theta$ is the Heaviside function. Results from combining an analysis of -fluctuations on the surface of critical droplets with \textsc{rg} recursion -relations suggest that $b=-(d-3)d/2$ for $d=2,4$ and $b=-7/3$ for $d=3$ -\cite{houghton.1980.metastable,rudnick.1976.equations,gunther.1980.goldstone}. -Assuming that $F$ is analytic in the upper complex-$h$ plane, the real part of -$F$ in the equilibrium state can be extracted from this imaginary metastable -free energy using the Kramers--Kronig relation -\[ - \re F(t,h)=\frac1\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{h'-h}\,\dd h'. - \label{eq:kram-kron} -\] -This relationship has been used to compute high-order moments of the free -energy with $H$ in good agreement with transfer matrix expansions -\cite{lowe.1980.instantons}. Here, we evaluate the integral explicitly to come -to -functional forms. In \threedee\ and \fourdee\ this can be done directly -given our scaling ansatz, yielding -\def\eqthreedeeone{ - \fF^\threedee(Y/B)&= - \frac{A}{12}\frac{e^{-1/Y^2}}{Y^2} - \bigg[\Gamma(\tfrac16)E_{7/6}(-Y^{-2}) -} -\def\eqthreedeetwo{ - -4Y\Gamma(\tfrac23)E_{5/3}(-Y^{-2})\bigg] -} -\def\eqfourdeeone{ - \fF^\fourdee(Y/B)&= - -\frac{A}{9\pi}\frac{e^{1/Y^3}}{Y^2} - \Big[3\ei(-Y^{-3}) -} -\def\eqfourdeetwo{ - +3\Gamma(\tfrac23)\Gamma(\tfrac13,Y^{-3}) - +\Gamma(\tfrac13)\Gamma(-\tfrac13,Y^{-3})\Big] -} -\ifreprint -\begin{align} - &\begin{aligned} - \eqthreedeeone\\ - &\hspace{7em} - \eqthreedeetwo - \end{aligned} - \\ - &\begin{aligned} - \eqfourdeeone - \\ - &\hspace{-0.5em} - \eqfourdeetwo, - \end{aligned} -\end{align} -\else -\begin{align} - \eqthreedeeone\eqthreedeetwo - \\ - \eqfourdeeone\eqfourdeetwo, -\end{align} -\fi -where $E_n$ is the generalized exponential integral and $\Gamma(x,y)$ is the -incomplete gamma function. -At the level of truncation of \eqref{eq:im.scaling} at which we are working -the Kramers--Kronig relation does not converge in \twodee. However, higher -moments can still be extracted, e.g., the susceptibility, by taking -\[ - \chi=\pd MH=-\frac1{T}\pd[2]Fh - =-\frac2{\pi T}\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. -\] -With a scaling form defined by $T\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, -this yields -\[ - \fX^\twodee(Y/B)=\frac{AB^2}{\pi Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big] - \label{eq:sus_scaling} -\] -Scaling forms for the free energy can then be extracted by direct integration -and their constants of integration fixed by known zero-field values, yielding -\begin{align} - \label{eq:mag_scaling} - \fM^\twodee(Y/B) - &=\fM(0)+\frac{AB}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\ - \fF^\twodee(Y/B) - &=-Y\bigg(\frac{\fM(0)}B-\frac{A}\pi e^{1/Y}\ei(-1/Y)\bigg) - \label{eq:2d_free_scaling} -\end{align} -with $F(t,h)=t^2\fF(h|t|^{-15/8})+t^2\log t^2$ in -two dimensions, as $\alpha=0$ and $\beta\delta=\frac{15}8$. - -How are these functional forms to be interpreted? Though the scaling function -\eqref{eq:im.scaling} for the imaginary free energy of the metastable state is -asymptotically correct sufficiently close to the critical point, the results -of the integral relation \eqref{eq:kram-kron} are not, since there is no limit -of $t$ or $h$ in which it becomes arbitrarily correct for a given truncation -of \eqref{eq:im.scaling}. It is well established that this method of using -unphysical or metastable elements of a theory to extract properties of the -stable or equilibrium theory is only accurate for high moments of those -predictions \cite{parisi.1977.asymptotic,bogomolny.1977.dispersion}. The -functions above should be understood as possessing exactly the correct -singularity at the coexistence line, but requiring polynomial corrections, -especially for smaller integer powers. Using these forms in conjunction with -existing methods of describing the critical equation of state or critical -properties with analytic functions in $h$ will incorporate these low-order -corrections while preserving the correct singular structure. In other words, -the scaling functions can be \emph{exactly} described by -$\tilde\fF(X)=\fF(X)+f(X)$ for some analytic function $f$. Higher order terms -in the expansion of $\tilde\fF$ become asymptotically equal to those of $\fF$ -because, as an analytic function, progressively higher order terms of $f$ must -eventually become arbitrarily small \cite{flanigan.1972.complex}. - -How predictive are these scaling forms in the proximity of the critical point -and the abrupt transition line? We simulated the \twodee\ Ising model on square -lattice using a form of the Wolff algorithm modified to remain efficient in -the presence of an external field. Briefly, the external field $H$ is applied -by adding an extra spin $s_0$ with coupling $|H|$ to all others -\cite{dimitrovic.1991.finite}. A quickly converging estimate for the -magnetization in the finite-size system was then made by taking -$M=\sgn(H)s_0\sum s_i$, i.e., the magnetization relative to the external spin -\cite{kent-dobias.2018.wolff}. -Data was then taken for susceptibility and magnetization for -$T_\c-T,H\leq0.1$. This data, rescaled as appropriate to collapse onto a -single curve, is plotted in Fig.~\ref{fig:scaling_fits}. - -For the \twodee\ Ising model on a square lattice, exact results at zero -temperature have $\fS(0)=4/T_\c$, $\fM(0)=(2^{5/2}\arcsinh1)^\beta$ -\cite{onsager.1944.crystal}, and $\fX(0)=C_0^-=0.025\,536\,971\,9$ \cite{barouch.1973.susceptibility}, so that -$B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$. If we -assume incorrectly that \eqref{eq:sus_scaling} is the true asymptotic form of -the susceptibility scaling function, then -$T\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ and the constant -$A$ is fixed to $A=\pi\fX(0)/2B^2=2^{19/8}\pi^3(\arcsinh1)^{15/4}C_0^-$. The -resulting scaling functions $\fX$ and $\fM$ are plotted as solid blue lines in -Fig.~\ref{fig:scaling_fits}. Though there is good agreement -between our functional forms and what is measured, there -are systematic differences that can be seen most clearly in the -magnetization. This is to be expected based on our earlier discussion: these -scaling forms should only be expected to well-describe the singularity at the -abrupt transition. Our forms both exhibit incorrect low-order -coefficients at the transition (Fig.~\ref{fig:series}) and incorrect -asymptotics as $h|t|^{-\beta\delta}$ becomes very large. - -In forthcoming work, -we develop a method to incorporate the essential singularity in the scaling -functions into a form that also incorporates known properties of the scaling -functions in the rest of the configuration space using a Schofield-like -parameterization \cite{schofield.1969.parametric,caselle.2001.critical,kent-dobias.2018.parametric}. Briefly, we define -parameters $R$ and $\theta$ by -\begin{align} - t=R(1-\theta^2) - && - h=h_0R^{\beta\delta}g(\theta) -\end{align} -where $h_0$ is a constant and $f$ is an arbitrary odd function whose first -finite zero $\theta_\c>1$ corresponds to the abrupt transition. In these -coordinates the invariant combination $h|t|^{-\beta\delta}$ is given by -\[ - h|t|^{-\beta\delta}=\frac{h_0g(\theta)}{|1-\theta^2|^{\beta\delta}}=\frac{h_0(-g'(\theta_\c))}{(\theta_c^2-1)^{\beta\delta}}(\theta_\c-\theta) - +\O\big( (\theta_\c-\theta)^2\big), -\] -an analytic function of $\theta$ about $\theta_\c$. -The simplest -function of the coordinate $\theta$ that exhibits the correct singularity at -the abrupt transitions at $h=\pm0$, $t<0$ is -\[ - \fX(\theta)=\fX\bigg(\frac{h_0(-g'(\theta_\c))}{(\theta_\c^2-1)^{\beta\delta}}(\theta_\c-\theta)\bigg)+\fX\bigg(\frac{h_0(-g'(\theta_\c))}{(\theta_\c^2-1)^{\beta\delta}}(\theta_\c+\theta)\bigg) -\] -This function is analytic in the range $-\theta_\c<\theta<\theta_\c$. -In order to correct its low-order behavior to match that expected, we -make use of both the freedom of the coordinate transformation $g$ and an -arbitrary analytic additive function $Y$, -\begin{align} - g(\theta)=\bigg(1-\frac{\theta^2}{\theta_\c^2}\bigg)\sum_{n=0}^\infty - g_n\theta^{2n+1} - && - Y(\theta)=\sum_{n=0}^\infty Y_n\theta^{2n} -\end{align} -so that $\tilde\fX(\theta)=\fX(\theta)+Y(\theta)$. By manipulating these -coefficients, we can attempt to give the resulting scaling form a series -expansion consistent with known values. One such prediction---made by fixing -the first four terms in the low-temperature, critical isotherm, and -high-temperature expansions -of $\tilde\fX$---is shown as a dashed yellow line in -Fig.~\ref{fig:scaling_fits}. -As shown in Fig.~\ref{fig:series}, the low-order free energy coefficients of this prediction match known values -exactly up to $n=5$, and improve the agreement with higher-order coefficients. -Unlike scaling forms which treat $\fX$ as analytic at the coexistence line, -the series coefficients of the scaling form developed here increase without -bound at high order. - - -\begin{figure} - \input{fig-susmag} - \caption{ - Scaling functions for (top) the susceptibility and (bottom) the - magnetization plotted in terms of the invariant combination - $h|t|^{-\beta\delta}$. Points with error bars show data with sampling - error taken from simulations of a $4096\times4096$ square-lattice Ising - model with periodic boundary conditions and $T_\c-T=0.01,0.02,\ldots,0.1$ - and $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. The solid blue lines - show our analytic results \eqref{eq:sus_scaling} and - \eqref{eq:mag_scaling}, the dashed yellow lines show - a scaling function modified to match known series expansions of the - susceptibility - to third order, and the - dotted green lines show the - polynomial resulting from truncating the known series expansion after the eight terms - reported by \cite{mangazeev.2008.variational,mangazeev.2010.scaling}. - } - \label{fig:scaling_fits} -\end{figure} - -\begin{figure} - \input{fig-series} - \caption{ - The series coefficients defined by $\tilde\fF(X)=\sum_nf_nX^n$. The blue - pluses correspond to the scaling form \eqref{eq:2d_free_scaling}, the - yellow saltires correspond to a scaling function modified to match known - series expansions of the susceptibility to third order---and therefore - the free energy to fifth order---and the green - stars - correspond to the first eight coefficients from - \cite{mangazeev.2008.variational,mangazeev.2010.scaling}. The modified - scaling function and the known coefficients match exactly up to $n=5$. - } - \label{fig:series} -\end{figure} - -Abrupt phase transitions, such as the jump in magnetization in the Ising -model below $T_\c$, are known to imply essential singularities in the free -energy that are usually thought to be unobservable in practice. We have -argued that this essential singularity controls the universal scaling -behavior near continuous phase transitions, and have derived an explicit -analytical form for the singularity in the free energy, magnetization, -and susceptibility for the Ising model. We have developed a Wolff algorithm -for the Ising model in a field, and showed that incorporating our singularity -into the scaling function gives good convergence to the simulations in \twodee. - -Our results should allow improved high-precision functional forms for the free -energy~\cite{caselle.2001.critical}, and should have implications for the scaling -of correlation functions~\cite{chen.2013.universal,wu.1976.spin}. Our methods might be generalized -to predict similar singularities in systems where nucleation and metastability -are proximate to continuous phase transitions, such as 2D superfluid -transitions~\cite{ambegaokar.1978.dissipation,ambegaokar.1980.dynamics}, the melting of 2D crystals~\cite{dahm.1989.dynamics}, and -freezing transitions in glasses, spin glasses, and other disordered systems. - - -%We have used results from the properties of the metastable Ising ferromagnet -%and the analytic nature of the free energy to derive universal scaling -%functions for the free energy, and in \twodee the magnetization and -%susceptibility, in the limit of small $t<0$ and $h$. Because of an essential -%singularity in these functions at $h=0$---the abrupt transition line---their -%form cannot be brought into that of any regular function by analytic -%redefinition of control or thermodynamic variables. These predictions match -%the results of simulations well. Having demonstrated that the essential -%singularity in thermodynamic functions at the abrupt transition leads to -%observable scaling effects, we hope that these functional forms will be used in -%conjunction with traditional perturbation methods to better express the -%equation of state of the Ising model in the whole of its parameter space. - -\begin{acknowledgments} - The authors would like to thank Tom Lubensky, Andrea Liu, and Randy Kamien - for helpful conversations. The authors would also like to think Jacques Perk - for pointing us to several insightful studies. JPS thanks Jim Langer for past inspiration, - guidance, and encouragement. This work was supported by NSF grants - DMR-1312160 and DMR-1719490. -\end{acknowledgments} - -\bibliography{essential-ising} - -\end{document} - diff --git a/ising_scaling.bib b/ising_scaling.bib new file mode 100644 index 0000000..36fdbe9 --- /dev/null +++ b/ising_scaling.bib @@ -0,0 +1,57 @@ +@article{Butera_2011_Free, + author = {Butera, P. and Pernici, M.}, + title = {Free energy in a magnetic field and the universal scaling equation of state for the three-dimensional Ising model}, + journal = {Phys. Rev. B}, + publisher = {American Physical Society}, + year = {2011}, + month = {2}, + volume = {83}, + pages = {054433}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.83.054433}, + doi = {10.1103/PhysRevB.83.054433}, + issue = {5}, + numpages = {15} +} + +@article{Campostrini_2000_Critical, + author = {Campostrini, Massimo and Pelissetto, Andrea and Rossi, Paolo and Vicari, Ettore}, + title = {Critical equation of state of three-dimensional {$XY$} systems}, + journal = {Physical Review B}, + publisher = {American Physical Society ({APS})}, + year = {2000}, + month = {9}, + number = {9}, + volume = {62}, + pages = {5843--5854}, + url = {https://doi.org/10.1103%2Fphysrevb.62.5843}, + doi = {10.1103/physrevb.62.5843} +} + +@article{Mangazeev_2008_Variational, + author = {Mangazeev, Vladimir V and Batchelor, Murray T and Bazhanov, Vladimir V and Dudalev, Michael Yu}, + title = {Variational approach to the scaling function of the 2D Ising model in a magnetic field}, + journal = {Journal of Physics A: Mathematical and Theoretical}, + publisher = {IOP Publishing}, + year = {2008}, + month = {12}, + number = {4}, + volume = {42}, + pages = {042005}, + url = {https://doi.org/10.1088%2F1751-8113%2F42%2F4%2F042005}, + doi = {10.1088/1751-8113/42/4/042005} +} + +@article{Mangazeev_2010_Scaling, + author = {Mangazeev, Vladimir V. and Dudalev, Michael Yu. and Bazhanov, Vladimir V. and Batchelor, Murray T.}, + title = {Scaling and universality in the two-dimensional Ising model with a magnetic field}, + journal = {Physical Review E}, + publisher = {American Physical Society (APS)}, + year = {2010}, + month = {6}, + number = {6}, + volume = {81}, + url = {https://doi.org/10.1103%2Fphysreve.81.060103}, + doi = {10.1103/physreve.81.060103} +} + + diff --git a/ising_scaling.tex b/ising_scaling.tex new file mode 100644 index 0000000..b7b76b2 --- /dev/null +++ b/ising_scaling.tex @@ -0,0 +1,42 @@ +\documentclass[aps,prb,reprint,longbibliography,floatfix]{revtex4-2} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{newtxtext, newtxmath} +\usepackage[ + colorlinks=true, + urlcolor=purple, + citecolor=purple, + filecolor=purple, + linkcolor=purple +]{hyperref} % ref and cite links with pretty colors + +\usepackage{amsmath, graphicx, xcolor} + +\begin{document} + +\title{Essential Singularities in Universal Scaling Functions at the Ising Coexistence Line} +\author{Jaron Kent-Dobias} +\author{James P.~Sethna} +\affiliation{Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA} + +\date\today + +\begin{abstract} +\end{abstract} + +\maketitle + +\cite{Campostrini_2000_Critical} + +\begin{acknowledgments} + The authors would like to thank Tom Lubensky, Andrea Liu, and Randy Kamien + for helpful conversations. The authors would also like to think Jacques Perk + for pointing us to several insightful studies. JPS thanks Jim Langer for past + inspiration, guidance, and encouragement. This work was supported by NSF + grants DMR-1312160 and DMR-1719490. +\end{acknowledgments} + +\bibliography{ising_scaling} + +\end{document} -- cgit v1.2.3-70-g09d2 From 306e8a2e6006ec2d589ad292fe7becce01723d23 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 23 Dec 2020 11:14:15 +0100 Subject: Started work on new version. --- ising_scaling.tex | 91 ++++++++++++++++++++++++++++++++++++++++++++++++++++--- 1 file changed, 86 insertions(+), 5 deletions(-) diff --git a/ising_scaling.tex b/ising_scaling.tex index b7b76b2..e229b33 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -1,4 +1,10 @@ -\documentclass[aps,prb,reprint,longbibliography,floatfix]{revtex4-2} +\documentclass[ + aps, + prb, + reprint, + longbibliography, + floatfix +]{revtex4-2} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} @@ -9,14 +15,18 @@ citecolor=purple, filecolor=purple, linkcolor=purple -]{hyperref} % ref and cite links with pretty colors - -\usepackage{amsmath, graphicx, xcolor} +]{hyperref} +\usepackage{amsmath} +\usepackage{graphicx} +\usepackage{xcolor} \begin{document} -\title{Essential Singularities in Universal Scaling Functions at the Ising Coexistence Line} +\title{Smooth Ising universal scaling functions} + \author{Jaron Kent-Dobias} +\affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France} + \author{James P.~Sethna} \affiliation{Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA} @@ -29,6 +39,77 @@ \cite{Campostrini_2000_Critical} + +\section{The 2D Ising model} + +\subsection{Definition of functions} + +\begin{equation} \label{eq:free.energy.2d.low} + F(u_t, u_h) + = |u_t|^2\mathcal F_{\pm}(u_h|u_t|^{-\beta\delta}) + +\frac{u_t^2}{8\pi}\log u_t^2 +\end{equation} +where the functions $\mathcal F_\pm$ have expansions in nonnegative integer powers of their arguments. +\begin{equation} \label{eq:free.energy.2d.mid} + F(u_t, u_h) + = |u_h|^{2/\beta\delta}\mathcal F_0(u_t|u_h|^{-1/\beta\delta}) + +\frac{u_t^2}{8\pi}\log u_h^{2/\beta\delta} +\end{equation} +where the function $\mathcal F_0$ has a convergent expansion in nonnegative integer powers of its argument \footnote{ + To connect with Mangazeev and Fonseca, $\mathcal F_0(x)=\tilde\Phi(-x)=\Phi(-x)+(x^2/8\pi) \log x^2$ and $\mathcal F_\pm(x)=G_{\mathrm{high}/\mathrm{low}}(x)$. +}. + +\begin{align} + \label{eq:schofield.free.energy} + F(R, \theta) &= R^2\mathcal F(\theta) + t(\theta)^2\frac{R^2}{8\pi}\log R^2 \\ + \label{eq:schofield.temperature} + u_t(R, \theta) &= Rt(\theta) \\ + \label{eq:schofield.field} + u_h(R, \theta) &= R^{\beta\delta}h(\theta) +\end{align} +The scaling function $\mathcal F$ can be defined in terms of the more conventional ones above by substituting \eqref{eq:schofield.temperature} and \eqref{eq:schofield.field} into \eqref{eq:free.energy.2d.low} and \eqref{eq:free.energy.2d.mid}, yielding +\begin{equation} + \begin{aligned} + &\mathcal F(\theta) + =t(\theta)^2\mathcal F_\pm\left[h(\theta)|t(\theta)|^{-\beta\delta}\right] + +\frac{t(\theta)^2}{8\pi}\log t(\theta)^2 \\ + &=|h(\theta)|^{2/\beta\delta}\mathcal F_0\left[t(\theta)|h(\theta)|^{-1/\beta\delta}\right] + +\frac{t(\theta)^2}{8\pi}\log h(\theta)^{2/\beta\delta} + \end{aligned} +\end{equation} +We choose the functions $t$ and $h$ so as to ensure that $F$ has an integer power series in \emph{all} regimes. $t$ is an even function of $\theta$ with $t(0)=1$ and $t(1)=0$. $h$ is an odd function with $h(0)=h(\theta_c)=0$ for some $\theta_c>1$. + +\begin{align} + t(\theta)&=1-\theta^2 \\ + h^{(n)}(\theta)&=\left(1-\frac{\theta^2}{\theta_c^2}\right)\sum_{i=0}^nh_i\theta^{2i+1} +\end{align} + + +\begin{equation} + f(x)=\Theta(-x) |x| e^{-1/|x|} +\end{equation} +where $\Theta$ is the Heaviside function. + +\begin{equation} + \operatorname{Im}\mathcal F(\theta)=A\left\{f\left[b(\theta_c-\theta)\right]+f\left[b(\theta_c+\theta)\right]\right\} +\end{equation} + +\begin{equation} + \begin{aligned} + \operatorname{Re}\mathcal F(\theta) + &=G(\theta^2)-\frac{\theta^2}\pi\int d\vartheta\, \frac{\operatorname{Im}\mathcal F(\vartheta)}{\vartheta^2(\vartheta-\theta)} \\ + &=G(\theta^2)+\frac A\pi\left\{f[b(\theta_c-\theta)]+f[b(\theta_c+\theta)]\right\} + \end{aligned} +\end{equation} +for arbitrary analytic function $G$ and +\begin{equation} + f(x)=xe^{1/x}\operatorname{Ei}(-1/x) +\end{equation} + +\section{The 3D Ising model} + +\section{Outlook} + \begin{acknowledgments} The authors would like to thank Tom Lubensky, Andrea Liu, and Randy Kamien for helpful conversations. The authors would also like to think Jacques Perk -- cgit v1.2.3-70-g09d2 From 347400221af3cc4c5846f2f0d50aa21268bbc5d8 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Thu, 24 Dec 2020 11:55:09 +0100 Subject: A lot of writing. --- ising_scaling.bib | 97 ++++++++++++++++++++- ising_scaling.tex | 252 +++++++++++++++++++++++++++++++++++++++++++++++------- 2 files changed, 317 insertions(+), 32 deletions(-) diff --git a/ising_scaling.bib b/ising_scaling.bib index 36fdbe9..9f4f14d 100644 --- a/ising_scaling.bib +++ b/ising_scaling.bib @@ -27,9 +27,101 @@ doi = {10.1103/physrevb.62.5843} } +@article{Caselle_2001_The, + author = {Caselle, Michele and Hasenbusch, Martin and Pelissetto, Andrea and Vicari, Ettore}, + title = {The critical equation of state of the two-dimensional {Ising} model}, + journal = {Journal of Physics A: Mathematical and General}, + publisher = {IOP Publishing}, + year = {2001}, + month = {3}, + number = {14}, + volume = {34}, + pages = {2923--2948}, + url = {https://doi.org/10.1088%2F0305-4470%2F34%2F14%2F302}, + doi = {10.1088/0305-4470/34/14/302} +} + +@phdthesis{Clement_2019_Respect, + author = {Clement, Colin}, + title = {Respect Your Data: Topics in Inference and Modeling in Physics}, + journal = {ProQuest Dissertations and Theses}, + year = {2019}, + url = {https://search.proquest.com/docview/2362109908/abstract/DCDCEAA7A8354B46PQ/1}, + address = {Ann Arbor}, + isbn = {9781392437766}, + school = {Cornell University} +} + +@article{Fonseca_2003_Ising, + author = {Fonseca, P. and Zamolodchikov, A.}, + title = {Ising Field Theory in a Magnetic Field: Analytic Properties of the Free Energy}, + journal = {Journal of Statistical Physics}, + publisher = {Springer Science and Business Media LLC}, + year = {2003}, + number = {3/6}, + volume = {110}, + pages = {527--590}, + url = {https://doi.org/10.1023%2Fa%3A1022147532606}, + doi = {10.1023/a:1022147532606} +} + +@article{Guida_1997_3D, + author = {Guida, R. and Zinn-Justin, J.}, + title = {{3D} {Ising} model: the scaling equation of state}, + journal = {Nuclear Physics B}, + publisher = {Elsevier BV}, + year = {1997}, + month = {4}, + number = {3}, + volume = {489}, + pages = {626--652}, + url = {https://doi.org/10.1016%2Fs0550-3213%2896%2900704-3}, + doi = {10.1016/s0550-3213(96)00704-3} +} + +@article{Gunther_1980_Goldstone, + author = {Günther, N J and Wallace, D J and Nicole, D A}, + title = {{Goldstone} modes in vacuum decay and first-order phase transitions}, + journal = {Journal of Physics A: Mathematical and General}, + publisher = {IOP Publishing}, + year = {1980}, + month = {5}, + number = {5}, + volume = {13}, + pages = {1755--1767}, + url = {https://doi.org/10.1088%2F0305-4470%2F13%2F5%2F034}, + doi = {10.1088/0305-4470/13/5/034} +} + +@article{Houghton_1980_The, + author = {Houghton, A. and Lubensky, T. C.}, + title = {The metastable {Ising} magnet in a negative field}, + journal = {Physics Letters A}, + publisher = {Elsevier BV}, + year = {1980}, + month = {6}, + number = {6}, + volume = {77}, + pages = {479--480}, + url = {https://doi.org/10.1016%2F0375-9601%2880%2990546-0}, + doi = {10.1016/0375-9601(80)90546-0} +} + +@phdthesis{Kent-Dobias_2020_Novel, + author = {Kent-Dobias, Jaron Patrick}, + title = {Novel Critical Phenomena}, + publisher = {ProQuest Dissertations Publishing}, + year = {2020}, + month = {August}, + url = {https://search.proquest.com/pqdtglobal/docview/2449126045/abstract/DA3EA013F5F140D1PQ/1}, + address = {Ann Arbor}, + isbn = {9798672130477}, + school = {Cornell University} +} + @article{Mangazeev_2008_Variational, author = {Mangazeev, Vladimir V and Batchelor, Murray T and Bazhanov, Vladimir V and Dudalev, Michael Yu}, - title = {Variational approach to the scaling function of the 2D Ising model in a magnetic field}, + title = {Variational approach to the scaling function of the {2D} {Ising} model in a magnetic field}, journal = {Journal of Physics A: Mathematical and Theoretical}, publisher = {IOP Publishing}, year = {2008}, @@ -43,13 +135,14 @@ @article{Mangazeev_2010_Scaling, author = {Mangazeev, Vladimir V. and Dudalev, Michael Yu. and Bazhanov, Vladimir V. and Batchelor, Murray T.}, - title = {Scaling and universality in the two-dimensional Ising model with a magnetic field}, + title = {Scaling and universality in the two-dimensional {Ising} model with a magnetic field}, journal = {Physical Review E}, publisher = {American Physical Society (APS)}, year = {2010}, month = {6}, number = {6}, volume = {81}, + pages = {060103}, url = {https://doi.org/10.1103%2Fphysreve.81.060103}, doi = {10.1103/physreve.81.060103} } diff --git a/ising_scaling.tex b/ising_scaling.tex index e229b33..1bd9843 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -37,38 +37,196 @@ \maketitle -\cite{Campostrini_2000_Critical} +At continuous phase transitions the thermodynamic properties of physical +systems have singularities. Celebrated renormalization group analyses imply +that not only the principal divergence but also entire additive functions are +\emph{universal}, meaning that they will appear at any critical points that +connect phases of the same symmetries in the same spatial dimension. The study +of these universal functions is therefore doubly fruitful: it provides both a +description of the physical or model system at hand, and \emph{every other +system} whose symmetries, interaction range, and dimension puts it in the same +universality class. +The continuous phase transition in the two-dimensional Ising model is perhaps +the most well studied, and its universal thermodynamic functions have likewise +received the most attention. Precision numeric work both on the lattice +critical theory and on the ``Ising'' critical field theory (related by +universality) have yielded high-order polynomial expansions of those functions +in various limits, along with a comprehensive understanding of their analytic +properties and even their full form \cite{Fonseca_2003_Ising, Mangazeev_2008_Variational, Mangazeev_2010_Scaling}. In parallel, smooth approximations of the +Ising ``equation of state'' have produced convenient, evaluable, differentiable +empirical functions \cite{Guida_1997_3D, Campostrini_2000_Critical, Caselle_2001_The}. Despite being differentiable, these approximations become +increasingly poor when derivatives are taken due to the presence of a subtle +essential singularity that is previously unaccounted for. + +This paper attempts to find the best of both worlds: a smooth approximate +universal thermodynamic function that respects the global analyticity of the +Ising free energy, for both the two-dimensional Ising model (where much is +known) and the three-dimensional Ising model (where comparatively less is +known). First, parametric coordinates are introduced that remove unnecessary +nonanalyticities from the scaling function. Then the arbitrary analytic +functions that compose those coordinates are approximated by truncated +polynomials whose coefficients are fixed by matching the series expansions of +the universal function in three critical regimes: at no field and low +temperature, no field and high temperature, and along the critical isotherm. + +This paper is divided into four parts. First, general aspects of the problem +will be reviewed that are relevant in all dimensions. Then, the process +described above will be applied to the two- and three-dimensional Ising models. + +\section{General aspects} + +\subsection{Universal scaling functions} + +Renormalization group analysis of the Ising critical point indicates that the free energy per site $f$ may be written, as a function of the reduced temperature $t=(T-T_c)/T_c$ and external field $h=H/T$, +\begin{equation} + f(t,h)=g(t,h)+f_s(t,h) +\end{equation} +with $g$ a nonuniversal analytic function that depends entirely on the system +in question and $f_s$ a singular function. The singular part $f_s$ can be said +to be universal in the following sense: for any system that shares the +universality with the Ising model, if the near-identity smooth change of coordinates +$u_t(t, h)$ and $u_h(t,h)$ is made such that the flow equations for the new +coordinates are exactly linearized, e.g., +\begin{align} \label{eq:flow} + \frac{du_t}{d\ell}=\frac1\nu u_t + && + \frac{du_h}{d\ell}=\frac{\beta\delta}\nu u_h, +\end{align} +then $f_s(u_t, u_h)$ will be the same function, up to constant rescalings of +the free energy and the nonlinear scaling fields $u_t$ and $u_h$. In order to +fix this last degree of freedom, we adopt the convention established by +\textbf{[probably earlier than what I'm citing here]} +\cite{Fonseca_2003_Ising}. The dependence of the nonlinear scaling variables on +the parameters $t$ and $h$ is also system-dependent, and their form can be +found for common model systems (the square- and triangular-lattice Ising +models) in the literature \cite{Clement_2019_Respect}. + +With the flow equations \eqref{eq:flow} along with that for the free energy, +the form of $f_s$ is highly constrained, further reduced to a universal +\emph{scaling function} of a single variable $u_h|u_t|^{-\beta\delta}$ (or equivalently +$u_tu_h^{-1/\beta\delta}$) with multiplicative power laws in $u_t$ or $u_h$ and +(sometimes) simple additive singular functions of $u_t$ and $u_h$. The special +variables are known as scaling invariants, as they are invariant under the flow +\eqref{eq:flow}. Reasonable assumptions about the analyticity of the scaling +function of a single variable then fixes the principal singularity at the +critical point. + +\subsection{Essential singularities and droplets} + +Another, more subtle, singularity exists which cannot be captured by the +multiplicative factors or additive terms, residing instead inside the scaling +function itself. The origin can be schematically understood to arise from a +singularity that exists in the complex free energy of the metastable phase of +the model, suitably continued into the equilibrium phase. When the equilibrium +Ising model with positive magnetization is subjected to a small negative +magnetic field, its equilibrium state instantly becomes one with a negative +magnetization. However, under physical dynamics it takes time to arrive at this +state, which happens after a fluctuation containing a sufficiently large +equilibrium `bubble' occurs. + +The bulk of such a bubble of radius $R$ lowers the free energy by +$2M|H|V_dR^d$, where $d$ is the dimension of space, $M$ is the magnetization, +$H$ is the external field, and $V_d$ is the volume of a $d$-ball, but its +surface raises the free energy by $\sigma S_dR^{d-1}$, where $\sigma$ is the +surface tension between the stable--metastable interface and $S_d$ is the +volume of a $(d-1)$-sphere. The bubble is sufficiently large to decay +metastable state when the differential bulk savings outweigh the surface costs. + +This critical bubble occurs with free energy cost +\begin{equation} + \begin{aligned} + \Delta F_c + &\simeq\left(\frac{S_d\sigma}d\right)^d\left(\frac{d-1}{2V_dM|H|}\right)^{d-1} \\ + &\simeq T\left(\frac{S_d\mathcal S(0)}d\right)^d\left[\frac{2V_d\mathcal M(0)}{d-1}ht^{-\beta\delta}\right]^{-(d-1)} + \end{aligned} +\end{equation} +where $\mathcal S(0)$ and $\mathcal M(0)$ are the critical amplitudes for the +surface tension and magnetization, respectively \textbf{[find more standard +notation]} \cite{Kent-Dobias_2020_Novel}. +In the context of statistical mechanics, Langer demonstrated that the decay rate is asymptotically proportional to the imaginary part of the free energy in the metastable phase, with (assuming Arrhenius behavior) +\begin{equation} + \operatorname{Im}f\propto\Gamma\sim e^{-\beta\Delta F_c}=e^{-1/(B|h||t|^{-\beta\delta})^{d-1}} +\end{equation} +which can be more rigorously related in the context of quantum field theory. + +This is a singular contribution that depends principally on the scaling +invariant $ht^{-\beta\delta}\simeq u_h|u_t|^{-\beta\delta}$. It is therefore +suggestive that this should be considered a part of the singular free energy +$f_s$, and moreover part of the scaling function that composes it. We will therefore make the ansatz that +\begin{equation} + \operatorname{Im}\mathcal F_-(\xi)=A\Theta(-\xi)|\xi|^{-b}e^{-1/(B|\xi|)^{d-1}}\left(1+O(\xi)\right) +\end{equation} +\cite{Houghton_1980_The} +The exponent $b$ depends on dimension and can be found through a more careful +accounting of the entropy of long-wavelength fluctuations in the droplet +surface \cite{Gunther_1980_Goldstone}. +Kramers--Kronig type dispersion relations can then be used to recover the +singular part of the real scaling function from this asymptotic form. + +\subsection{Schofield coordinates} + +The invariant combinations $u_h|u_t|^{-\beta\delta}$ or +$u_t|u_h|^{-1/\beta\delta}$ are natural variables to describe the scaling +functions, but prove unwieldy when attempting to make smooth approximations. +This is because, when defined in terms of these variables, scaling functions +that have polynomial expansions at small argument have nonpolynomial expansions +at large argument. Rather than deal with the creative challenge of dreaming up +functions with different asymptotic expansions in different limits, we adopt +different coordinates, in terms of which a scaling function can be defined that +has polynomial expansions in \emph{all} limits. + +In all dimensions, the Schofield coordinates $R$ and $\theta$ will be implicitly defined by +\begin{align} \label{eq:schofield} + u_t(R, \theta) = Rt(\theta) + && + u_h(R, \theta) = R^{\beta\delta}h(\theta) +\end{align} +where $t$ and $h$ are polynomial functions selected so as to associate different scaling limits with different values of $\theta$. We will adopt standard forms for these functions, given by +\begin{align} \label{eq:schofield.funcs} + t(\theta)=1-\theta^2 + && + h(\theta)=\left(1-\frac{\theta^2}{\theta_c^2}\right)\sum_{i=0}^\infty h_i\theta^{2i+1} +\end{align} +This means that $\theta=0$ corresponds to the high-temperature zero-field line, +$\theta=1$ to the critical isotherm at nonzero field, and $\theta=\theta_c$ to +the low-temperature zero-field (phase coexistence) line. + +In practice the infinite series in \eqref{eq:schofield.funcs} cannot be +entirely fixed, and it will be truncated at finite order. We will notate the +truncation an upper bound of $n$ by $h^{(n)}$. The convergence of the +coefficients as $n$ is increased will be part of our assessment of the success +of the convergence of the scaling form. \section{The 2D Ising model} \subsection{Definition of functions} +The scaling function for the two-dimensional Ising model is the most +exhaustively studied universal forms in statistical physics and quantum field +theory. \begin{equation} \label{eq:free.energy.2d.low} - F(u_t, u_h) + f_s(u_t, u_h) = |u_t|^2\mathcal F_{\pm}(u_h|u_t|^{-\beta\delta}) +\frac{u_t^2}{8\pi}\log u_t^2 \end{equation} where the functions $\mathcal F_\pm$ have expansions in nonnegative integer powers of their arguments. \begin{equation} \label{eq:free.energy.2d.mid} - F(u_t, u_h) + f_s(u_t, u_h) = |u_h|^{2/\beta\delta}\mathcal F_0(u_t|u_h|^{-1/\beta\delta}) +\frac{u_t^2}{8\pi}\log u_h^{2/\beta\delta} \end{equation} -where the function $\mathcal F_0$ has a convergent expansion in nonnegative integer powers of its argument \footnote{ - To connect with Mangazeev and Fonseca, $\mathcal F_0(x)=\tilde\Phi(-x)=\Phi(-x)+(x^2/8\pi) \log x^2$ and $\mathcal F_\pm(x)=G_{\mathrm{high}/\mathrm{low}}(x)$. -}. - -\begin{align} - \label{eq:schofield.free.energy} - F(R, \theta) &= R^2\mathcal F(\theta) + t(\theta)^2\frac{R^2}{8\pi}\log R^2 \\ - \label{eq:schofield.temperature} - u_t(R, \theta) &= Rt(\theta) \\ - \label{eq:schofield.field} - u_h(R, \theta) &= R^{\beta\delta}h(\theta) -\end{align} -The scaling function $\mathcal F$ can be defined in terms of the more conventional ones above by substituting \eqref{eq:schofield.temperature} and \eqref{eq:schofield.field} into \eqref{eq:free.energy.2d.low} and \eqref{eq:free.energy.2d.mid}, yielding -\begin{equation} +where the function $\mathcal F_0$ has a convergent expansion in nonnegative integer powers of its argument. +To connect with Mangazeev and Fonseca, $\mathcal F_0(x)=\tilde\Phi(-x)=\Phi(-x)+(x^2/8\pi) \log x^2$ and $\mathcal F_\pm(x)=G_{\mathrm{high}/\mathrm{low}}(x)$. + +Schofield coordinates all us to define a global scaling function $\mathcal F$ by +\begin{equation} \label{eq:schofield.2d.free.energy} + f_s(R, \theta) = R^2\mathcal F(\theta) + t(\theta)^2\frac{R^2}{8\pi}\log R^2 +\end{equation} +The scaling function $\mathcal F$ can be defined in terms of the more +conventional ones above by substituting \eqref{eq:schofield} into \eqref{eq:free.energy.2d.low} and +\eqref{eq:free.energy.2d.mid}, yielding +\begin{equation} \label{eq:scaling.function.equivalences.2d} \begin{aligned} &\mathcal F(\theta) =t(\theta)^2\mathcal F_\pm\left[h(\theta)|t(\theta)|^{-\beta\delta}\right] @@ -77,39 +235,73 @@ The scaling function $\mathcal F$ can be defined in terms of the more convention +\frac{t(\theta)^2}{8\pi}\log h(\theta)^{2/\beta\delta} \end{aligned} \end{equation} -We choose the functions $t$ and $h$ so as to ensure that $F$ has an integer power series in \emph{all} regimes. $t$ is an even function of $\theta$ with $t(0)=1$ and $t(1)=0$. $h$ is an odd function with $h(0)=h(\theta_c)=0$ for some $\theta_c>1$. - -\begin{align} - t(\theta)&=1-\theta^2 \\ - h^{(n)}(\theta)&=\left(1-\frac{\theta^2}{\theta_c^2}\right)\sum_{i=0}^nh_i\theta^{2i+1} -\end{align} +Examination of \eqref{eq:scaling.function.equivalences.2d} finds that $\mathcal F$ has expansions in integer powers in the entire domain $-\theta_c\leq0\leq\theta_c$. -\begin{equation} +\begin{equation} \label{eq:im.f.func.2d} f(x)=\Theta(-x) |x| e^{-1/|x|} \end{equation} -where $\Theta$ is the Heaviside function. - \begin{equation} - \operatorname{Im}\mathcal F(\theta)=A\left\{f\left[b(\theta_c-\theta)\right]+f\left[b(\theta_c+\theta)\right]\right\} + \operatorname{Im}\mathcal F(\theta)=A\left\{f\left[\tilde B(\theta_c-\theta)\right]+f\left[b(\theta_c+\theta)\right]\right\} \end{equation} \begin{equation} \begin{aligned} \operatorname{Re}\mathcal F(\theta) &=G(\theta^2)-\frac{\theta^2}\pi\int d\vartheta\, \frac{\operatorname{Im}\mathcal F(\vartheta)}{\vartheta^2(\vartheta-\theta)} \\ - &=G(\theta^2)+\frac A\pi\left\{f[b(\theta_c-\theta)]+f[b(\theta_c+\theta)]\right\} + &=G(\theta^2)+\frac A\pi\left\{f[\tilde B(\theta_c-\theta)]+f[\tilde B(\theta_c+\theta)]\right\} \end{aligned} \end{equation} -for arbitrary analytic function $G$ and +for arbitrary analytic function $G$ with +\begin{equation} + G(x)=\sum_{i=0}^\infty G_ix^i +\end{equation} +and $f$ is \begin{equation} f(x)=xe^{1/x}\operatorname{Ei}(-1/x) \end{equation} +the Kramers--Kronig transformation of \eqref{eq:im.f.func.2d}, where $\operatorname{Ei}$ is the exponential integral. + +\subsection{Iterative fitting} + +\subsection{Comparison with other smooth forms} -\section{The 3D Ising model} +\section{The three-dimensional Ising model} + +The three-dimensional Ising model is easier in some ways, since its hyperbolic critical point lacks stray logarithms. + +\begin{equation} \label{eq:free.energy.3d.low} + f_s(u_t, u_h) + = |u_t|^{2-\alpha}\mathcal F_{\pm}(u_h|u_t|^{-\beta\delta}) +\end{equation} +\begin{equation} \label{eq:free.energy.3d.mid} + f_s(u_t, u_h) + = |u_h|^{(2-\alpha)/\beta\delta}\mathcal F_0(u_t|u_h|^{-1/\beta\delta}) +\end{equation} +\begin{equation} \label{eq:schofield.3d.free.energy} + f_s(R, \theta) = R^2\mathcal F(\theta) +\end{equation} +\begin{equation} \label{eq:scaling.function.equivalences.3d} + \begin{aligned} + \mathcal F(\theta) + &=t(\theta)^{2-\alpha}\mathcal F_\pm\left[h(\theta)|t(\theta)|^{-\beta\delta}\right] \\ + &=|h(\theta)|^{(2-\alpha)/\beta\delta}\mathcal F_0\left[t(\theta)|h(\theta)|^{-1/\beta\delta}\right] + \end{aligned} +\end{equation} +\begin{equation} \label{eq:im.f.func.3d} + f(x)=\Theta(-x) |x|^{-7/3} e^{-1/|x|^2} +\end{equation} +\begin{equation} + f(x)=\frac{e^{-1/x^2}}{12}\left[ + \frac4x\Gamma\big(\tfrac23\big)\operatorname{E}_{\frac53}(-x^{-2}) + -\frac1{x^2}\Gamma\big(\tfrac16\big)\operatorname{E}_{\frac76}(-x^{-2}) + \right] +\end{equation} \section{Outlook} +The successful smooth description of the Ising free energy produced in part by analytically continuing the singular imaginary part of the metastable free energy inspires an extension of this work: a smooth function that captures the universal scaling \emph{through the coexistence line and into the metastable phase}. Indeed, the tools exist to produce this: by writing $t(\theta)=(1-\theta^2)(1-(\theta/\theta_m)^2)$ for some $\theta_m>\theta_c$, the invariant scaling combination + \begin{acknowledgments} The authors would like to thank Tom Lubensky, Andrea Liu, and Randy Kamien for helpful conversations. The authors would also like to think Jacques Perk -- cgit v1.2.3-70-g09d2 From 9903d8a6639ce0bf6b3f8ec790366a036fe9032d Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 3 Feb 2021 13:16:03 +0100 Subject: Removed currently unused data. --- data/data_hex-2048.dat | 500 ---------------------------------------------- data/data_hex-4096.dat | 10 - data/data_square-1024.dat | 484 -------------------------------------------- data/data_square-2048.dat | 500 ---------------------------------------------- data/data_square-4096.dat | 500 ---------------------------------------------- data/data_tri-2048.dat | 500 ---------------------------------------------- data/data_tri-4096.dat | 41 ---- 7 files changed, 2535 deletions(-) delete mode 100644 data/data_hex-2048.dat delete mode 100644 data/data_hex-4096.dat delete mode 100644 data/data_square-1024.dat delete mode 100644 data/data_square-2048.dat delete mode 100644 data/data_square-4096.dat delete mode 100644 data/data_tri-2048.dat 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